Euler systems for Rankin--Selberg convolutions of modular forms
aa r X i v : . [ m a t h . N T ] A ug EULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULARFORMS
ANTONIO LEI, DAVID LOEFFLER, AND SARAH LIVIA ZERBES
Abstract.
We construct an Euler system in the cohomology of the tensor product of the Galoisrepresentations attached to two modular forms, using elements in the higher Chow groups of productsof modular curves. We use these elements to prove a finiteness theorem for the strict Selmer groupof the Galois representation when the associated p -adic Rankin–Selberg L -function is non-vanishing at s = 1. Dedicated to Kazuya Kato
Contents
1. Outline 12. Generalized Beilinson–Flach elements 23. Norm relations for generalized Beilinson–Flach elements 144. Relation to complex L-values 375. Relation to p -adic L -values 436. Families of cohomology classes 497. Bounding strict Selmer groups 628. Conjectures on higher-rank Euler systems 71Appendix A. Ancillary results 74References 771. Outline
In [BDR12], Bertolini, Darmon and Rotger have studied certain canonical global cohomology classes(the “Beilinson–Flach elements”, obtained from the constructions of [Be˘ı84] and [Fla92]) in the coho-mology of the tensor products of the p -adic Galois representations of pairs of weight 2 modular forms,and related their image under the Bloch–Kato logarithm maps to the values of p -adic Rankin–Selberg L -functions. These Beilinson–Flach elements are constructed as the image of elements of the higherChow group of a product of modular curves.In this paper, we construct a form of Euler system – a compatible system of cohomology classes overcyclotomic fields – of which the Beilinson–Flach elements are the bottom layer. We first define elementsof higher Chow groups of the product of two (affine) modular curves over a cyclotomic field, c Ξ m,N,j ∈ CH ( Y ( N ) ⊗ Q ( µ m ) , m ≥ N ≥
5, and j ∈ Z /m Z (cf. Definition 2.7.3). These are obtained by considering theimages of various maps from higher level modular curves to the surface Y ( N ) , together with modularunits (Siegel units) on these curves. For m = 1 our elements reduce to those considered in [BDR12],and as in op.cit. , we show that after tensoring with Q we can construct preimages of our elements in thehigher Chow group of the self-product of the projective modular curve X ( N ); however, in this paper(as in [Kat04]) we shall take the affine versions as the principal objects of study. Date : Submitted February 7, 2013; revised August 23, 2013.The authors’ research is supported by the following grants: CRM-ISM Postdoctoral Fellowship (Lei); Royal SocietyUniversity Research Fellowship (Loeffler); EPSRC First Grant EP/J018716/1 (Zerbes).
We show two forms of compatibility relation for our generalized Beilinson–Flach elements: firstly,relating c Ξ m,N,j to the pushforward of c Ξ m,Np,j , for p prime (Theorem 3.1.2); secondly, relating c Ξ m,N,j to the pushforward (or Galois norm) of c Ξ mp,N,j , where p is prime and either p | N (Theorem 3.3.2) or p ∤ mN (Theorem 3.4.1).We next turn to the relation between our elements and L -values. Theorem 4.3.7 shows, following anargument due to Beilinson, that the images of the elements c Ξ m,N,j under the Beilinson regulator mapinto complex de Rham cohomology are related to the derivatives at s = 1 of Rankin–Selberg L -functionsof weight 2 modular forms. Theorem 5.6.4 is a p -adic analogue of this result, generalizing a theorem ofBertolini–Darmon–Rotger [BDR12]; it gives a formula for the image of our element for m = 1 under the p -adic syntomic regulator, for a prime p ∤ N , in terms of Hida’s p -adic Rankin–Selberg L -functions.Next we consider the images of our elements in ´etale cohomology. Applying Huber’s “continuous ´etalerealization” functor and the Hochschild–Serre exact sequence, and projecting into the isotypical compo-nent corresponding to a pair of eigenforms ( f, g ) of level N , allows us to construct Galois cohomologyclasses c z ( f,g,N ) m ∈ H ( Q ( µ m ) , V ∗ f ⊗ V ∗ g )from the elements c Ξ m,N,j ; see Definition 6.4.4. Using the second norm relation in the p -adic cyclotomictower, we can modify these to construct elements of Iwasawa cohomology groups of pairs of modularforms (under a strong “ordinarity” hypothesis; this is Theorem 6.8.6), or of the tensor product of Iwasawacohomology groups with the algebra of distributions (under a weaker “small slope” hypothesis; seeTheorem 6.8.4). These elements satisfy compatibility relations of Euler-system type when additionalprimes are added to m .Using the first norm relation, we also obtain variation in Hida families. More specifically, if one of thetwo forms (say g ) is ordinary, we may deform our cohomology classes analytically as g varies over a Hidafamily; cf. Theorem 6.9.5. (In the special case m = 1, such results have been independently obtained byBertolini–Darmon–Rotger.) When f and g are both ordinary, we obtain three-variable families whichalso incorporate cyclotomic twists, cf. Theorem 6.9.8.As an application of these constructions, we prove (under some technical hypotheses) a finitenesstheorem for the strict Selmer group of a product of modular forms (Theorem 7.4.2) when the associated p -adic Rankin–Selberg L -function is non-vanishing at s = 1, and (under very slightly stronger hypotheses)we give an explicit bound for the order of the strict Selmer group in terms of the p -adic L -value (Theorem7.5.1). Remark.
It is a pleasure to acknowledge the very deep debt this article owes to the magisterial workof Kato [Kat04]. We have adopted many aspects of the strategy and methods of Kato’s work, reuseda number of his results, and even in many cases adopted his notation. It is a pleasure to dedicate thiswork to Professor Kato, as a humble gift on the occasion of his 60th birthday.
Acknowledgements.
Part of this work was done while the second and the third author were visitingMontr´eal and Bielefeld in Spring 2012; they would like to thank Henri Darmon and Thomas Zink for theirhospitality. We would also like to thank Massimo Bertolini, Francois Brunault, Francesc Castella, JohnCoates, Henri Darmon, Mathias Flach, Kazuya Kato, Masato Kurihara, Andreas Langer, Jan Nekovar,Ken Ribet, Victor Rotger, Karl Rubin, Tony Scholl, Xin Wan and Andrew Wiles for helpful comments.We are also grateful to the anonymous referee for a number of valuable suggestions and corrections.2.
Generalized Beilinson–Flach elements
In this section we shall construct elements of motivic cohomology groups of products of modularcurves, using the explicit description of the motivic cohomology given by the Gersten complex.2.1.
Modular curves.
We begin by fixing notation regarding modular curves. We follow the conven-tions of [Kat04] very closely (see in particular §§
1, 2, and 5 of op.cit. ). Definition 2.1.1.
For N ≥ , let Y ( N ) denote the smooth affine curve over Q which represents thefunctor on the category of Q -schemes S isomorphism classes of triples ( E, e , e ) , E an elliptic curve over S , e , e sections of E/S generating E [ N ] . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 3
The variety Y ( N ) is equipped with a left action of GL ( Z /N Z ): the element (cid:18) a bc d (cid:19) maps ( E, e , e )to ( E, e ′ , e ′ ) where (cid:18) e ′ e ′ (cid:19) = (cid:18) a bc d (cid:19) (cid:18) e e (cid:19) . In particular, this action factors through GL ( Z /N Z ) / h± i .There is an obvious surjective morphism Y ( N ) → µ ◦ N , where µ ◦ N is the scheme of primitive N -throots of unity, which sends ( E, e , e ) to h e , e i E [ N ] , where h− , −i E [ N ] denotes the Weil pairing on E [ N ]. Because the Weil pairing is non-degenerate and alternating, the induced action of GL ( Z /N Z )on µ ◦ N is given by σ · ζ = ζ det σ ; and the fibre of Y ( N )( C ) over the point e πi/N ∈ µ ◦ N ( C ) is canonicallyand SL ( Z /N Z )-equivariantly identified with Γ( N ) \H , where H is the upper half-plane and Γ( N ) theprincipal congruence subgroup of level N in SL ( Z ), via the map τ ( C / ( Z + Z τ ) , τ /N, /N ) . We shall mainly be working with certain quotients of the curves Y ( N ), which we now define. Definition 2.1.2.
For
M, N ≥ , we shall define Y ( M, N ) to be the quotient of Y ( L ) , for any L ≥ divisible by M and N , by the group (cid:26)(cid:18) a bc d (cid:19) ∈ GL ( Z /L Z ) : a = 1 , b = 0 mod M,c = 0 , d = 1 mod N (cid:27) . This curve Y ( M, N ) represents the functor of triples (
E, e , e ) where e has order M , e has order N , and e , e generate a subgroup of E of order M N . Definition 2.1.3.
We write Y ( N ) for the smooth affine curve over Q representing the functor S isomorphism classes of pairs ( E, P ) , E an elliptic curve over S , P a section of E/S of exact order N . Remark . Note that the cusp ∞ , which corresponds to the generalized elliptic curve (cid:0) G m /q Z , ζ N (cid:1) ,is not defined over Q [[ q ]] but rather over Q ( µ N )[[ q ]], so the q -expansions of elements of O ( Y ( N )) do notnecessarily lie in Q (( q )) but rather in Q ( µ N )(( q )). See e.g. [DI95, § Y ( N ) = Y (1 , N ). More generally, we may use the following proposition to identify Y ( N ) × µ ◦ m , for m, N ≥
1, with a quotient of a principal modular curve:
Proposition 2.1.5. If N ≥ , m ≥ , and L ≥ is divisible by both N and m , then the map Y ( L ) ✲ Y ( N ) ⊗ µ ◦ m ( E, e , e ) ✲ h(cid:0) E, LN e (cid:1) , (cid:10) Lm e , Lm e (cid:11) E [ m ] i identifies Y ( N ) × µ ◦ m with the quotient of Y ( L ) by the subgroup of GL ( Z /L Z ) given by (cid:26)(cid:18) a bc d (cid:19) : c = 0 , d = 1 mod N,ad − bc = 1 mod m (cid:27) . We shall be most interested in the curves Y ( m, mN ) for m ≥ , N ≥
1. Note that Y ( m, mN ) mapsnaturally to µ ◦ m , with geometrically connected fibres. It has a left action of the group (cid:26)(cid:18) a bc d (cid:19) : c = 0 mod N (cid:27) , compatible with the determinant action on µ ◦ m : if x = ( E, e , e ) is a point of Y ( m, mN ), so e has order m and e has order mN , and g = (cid:18) a bc d (cid:19) ∈ GL ( Z /mN Z ) with N | c , then g · x = ( E, ae + bN e , c/N e + de ) . We shall introduce some notation for maps between these curves.
Definition 2.1.6.
Let m, N ≥ .(1) We write t m for the morphism Y ( m, mN ) → Y ( N ) × µ ◦ m given by ( E, e , e ) (cid:2) ( E/ h e i , [ me ]) , h e , N e i E [ m ] (cid:3) . A. LEI, D. LOEFFLER, AND S.L. ZERBES (2) For a ≥ , we write τ a for the morphism Y ( am, amN ) → Y ( m, mN ) given by ( E, e , e ) ( E/C, [ e ] , [ ae ]) where C is the cyclic subgroup of order a generated by me , and [ e ] , [ ae ] denote the images of e and ae on E/C . Proposition 2.1.7.
Let m, N, a as above.(1) We have a commutative diagram Y ( am, amN ) τ a ✲ Y ( m, mN ) Y ( N ) × µ ◦ am t am ❄ ✲ Y ( N ) × µ ◦ m ,t m ❄ where the bottom horizontal arrow is the identity map on Y ( N ) and the map µ ◦ am → µ ◦ m givenby ζ ζ a .(2) For b ∈ ( Z /mN Z ) × , the map t m intertwines the action of ( b
00 1 ) with the automorphism σ b : ζ ζ b of µ ◦ m , and (cid:0) b − b (cid:1) with the diamond operator h b i on Y ( N ) .Proof. The first statement is immediate from the definition of the maps and properties of the Weilpairing, and the second is an easy verification (cf. [Kat04, 5.7.1]). (cid:3)
Remark . The use of the maps τ a is forced on us by the nature of our construction of zeta elements.It would be much more satisfying to use the natural degeneracy maps Y ( am, amN ) → Y ( m, mN ) givenby τ ′ a : ( E, e , e ) ( E, ae , ae ), but we do not know how to construct elements compatible under thesemaps; see § Notation.
For compatibility with [Kat04], we shall use the alternative notation Y ( N ) ⊗ Q ( µ m ) for Y ( N ) × µ ◦ m .We shall also have to deal with products of two modular curves. Definition 2.1.9.
We shall write Y ( N ) (slightly abusively) for the fibre product Y ( N ) × µ ◦ N Y ( N ) . Thisis a subvariety of Y ( N ) × Spec( Q ) Y ( N ) preserved by the subgroup (cid:8) ( σ, τ ) ∈ GL ( Z /N Z ) : det( σ ) = det( τ ) (cid:9) . Similarly, we shall write Y ( m, mN ) for Y ( m, mN ) × µ ◦ m Y ( m, mN ) , which is acted upon by the group G = (cid:26) ( σ, τ ) ∈ GL ( Z /mN Z ) : det( σ ) = det( τ ) mod m, σ, τ = (cid:18) ∗ ∗ ∗ (cid:19) mod N. (cid:27) Evidently, the image of Y ( m, mN ) × µ ◦ m Y ( m, mN ) under t m × t m lands in( Y ( N ) × µ ◦ m ) × µ ◦ m ( Y ( N ) × µ ◦ m ) = Y ( N ) × µ ◦ m , so we may consider t m × t m as a morphism Y ( m, mN ) → Y ( N ) × µ ◦ m , which intertwines the action of(( b
00 1 ) , ( b
00 1 )) ∈ G with σ b .We shall also have to consider, occasionally, some more general classes of modular curves. Here weshall only consider models over Q . Definition 2.1.10. If Γ ⊆ SL ( Z ) is a congruence subgroup, then we shall write Y (Γ) for the variety (Γ \ Y ( L )) ⊗ Q ( µ L ) Q , where L is any integer ≥ such that Γ ⊇ Γ( L ) ; this variety is independent of the choice of L .Remark . If α ∈ GL +2 ( Q ), then the isomorphism of Riemann surfaces Y (Γ)( C ) ∼ = Y ( α Γ α − )( C )mapping τ ∈ H to ατ extends to an algebraic isomorphism defined over Q ; similarly for degeneracy maps Y (Γ) → Y (Γ ′ ) for Γ ⊆ Γ ′ .The above constructions with affine modular curves also have projective analogues where the cusps aretaken into account; we shall write X ( − ) for the compactified version of Y ( − ) in line with the standardnotation. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 5
Siegel units.Definition 2.2.1.
For ( α, β ) ∈ ( Q / Z ) − { (0 , } of order dividing N , and c > coprime to N , let c g α,β ∈ O ( Y ( N )) × denote Kato’s Siegel unit, as defined in [Kat04, § . We identify c g α,β with a holomorphic function on the upper half-plane, via the identification of thefibre of Y ( N )( C ) over e πi/N ∈ µ ◦ N ( C ) with Γ( N ) \H given in the previous section. (Note that c g α,β isdefined over Q as a function on Y ( N ), but in order to interpret it as a holomorphic function on H wemust make a choice of N -th root of unity, and the q -expansion coefficients of c g α,β are in Q ( µ N ).)Recall that there is an element g α,β ∈ O ( Y ( N )) × ⊗ Q such that c g α,β = c g α,β − g cα,cβ . Proposition 2.2.2 (Distribution relations) . Let m ≥ , and let c be a nonzero integer coprime to m and the orders of α, β . Then the following relations hold: (1a) c g α,β ( mz ) = Y β ′ c g α,β ′ ( z ) where the product is over β ′ ∈ Q / Z such that mβ ′ = β ; (1b) c g α,β ( z/m ) = Y α ′ c g α ′ ,β ( z ) where the product is over α ′ ∈ Q / Z such that mα ′ = α ; and (1c) c g α,β ( z ) = Y α ′ ,β ′ c g α ′ ,β ′ ( z ) where the product is over pairs ( α ′ , β ′ ) ∈ ( Q / Z ) such that ( mα ′ , mβ ′ ) = ( α, β ) .Proof. Formula (1a) is Lemma 2.12 of [Kat04]. Formula (1b) can be proved similarly, or can be deduceddirectly from (1a) using the action of (cid:18) −
11 0 (cid:19) . Formula (1c), which is Lemma 1.7(2) of op.cit. , isimmediate by combining (1) and (2). (cid:3)
Remark . The three formulae above admit the following common generalization: let M be a 2 × D . Then we have c g α,β ( M · z ) = Y α ′ ,β ′ c g α ′ ,β ′ where the product is over all ( α ′ , β ′ ) such that ( α ′ , β ′ ) M ′ = ( α, β ), where M ′ = (det M ) M − is theadjugate matrix of M . Cases (1), (2) and (3) correspond to taking M = (cid:18) m
00 1 (cid:19) , (cid:18) m (cid:19) and (cid:18) m m (cid:19) respectively. The case where M is invertible is (part of) Lemma 1.7(1) of op.cit. .We are most interested in the units c g , /N , which descend to units on Y ( N ). These have the followingcompatibility property: Theorem 2.2.4 (Kato) . If M, N, N ′ ≥ are integers with prime( N ′ ) = prime( N ) , and α is the naturalprojection Y ( M, N ′ ) → Y ( M, N ) (which induces a norm map α ∗ : O ( Y ( M, N ′ )) × → O ( Y ( M, N )) × ),then we have α ∗ ( c g , /N ′ ) = c g , /N . If N ′ = N ℓ , where ℓ is prime and ℓ ∤ M N , then we have α ∗ ( c g , /N ′ ) = c g , /N · (cid:0) c g , “ ℓ − ” /N (cid:1) − where “ ℓ − ” signifies the inverse of ℓ modulo N .Proof. These statements are proved in [Kat04, § K , Propositions 2.3 and 2.4 of op.cit. ). We reproduce the proofs brieflyhere. A. LEI, D. LOEFFLER, AND S.L. ZERBES
Firstly, let us suppose prime( mN ′ ) = prime( mN ). Let a = N ′ /N . Since prime( mN ′ ) = prime( mN ),for each ( x, y ) ∈ ( Z /a Z ) we may choose an element s xy ∈ GL ( Z /mN ′ Z ) of the form (cid:18) mN x mN y (cid:19) . These elements s xy are coset representatives for the quotient of the two subgroups of GL ( Z /mN ′ Z )corresponding to Y ( m, mN ) and Y ( m, mN ′ ), so we have α ∗ (cid:0) c g , /N ′ (cid:1) = Y x,y s ∗ xy (cid:0) c g , /N ′ (cid:1) . For any M ≥
1, any u ∈ GL ( Z /M Z ) and any α, β ∈ (cid:0) M Z / Z (cid:1) , we have u ∗ ( c g α,β ) = c g α ′ ,β ′ where ( α ′ , β ′ ) = ( α, β ) · u ;applying this to the formula above we deduce that α ∗ (cid:0) c g , /mN ′ (cid:1) = Y x,y ∈ Z /a Z (cid:0) c g x/a, /mN ′ + y/a (cid:1) . The latter expression is equal to the product of c g γ,δ over all pairs ( γ, δ ) such that ( aγ, aδ ) = (0 , /mN );so using the distribution property of Equation (1a), the product is c g , /mN as required.In the second case, where N ′ = ℓN for ℓ ∤ M N , we pass via the intermediate modular curves Y ( M, N ( ℓ )) and Y ( M ( ℓ ) , N ) described in [Kat04, § ϕ ℓ : Y ( M, N ( ℓ )) → Y ( M ( ℓ ) , N ) be the mapdefined in op.cit. , corresponding to z ℓz on H . We factor the projection α as α ◦ α , where α and α are the natural maps Y ( M, N ℓ ) α ✲ Y ( M, N ( ℓ )) α ✲ Y ( M, N ) . By [Kat04, Step 2 of § α ) ∗ (cid:0) c g , /Nℓ (cid:1) = ϕ ∗ ℓ (cid:0) c g , /N (cid:1) · (cid:0) c g , “ ℓ − ” /N (cid:1) − ;( α ) ∗ ϕ ∗ ℓ ( c g , /N ) = c g , /N · ( c g , “ ℓ − ” /N ) ℓ ;( α ) ∗ (cid:0) c g , “ ℓ − ” /N (cid:1) = (cid:0) c g , “ ℓ − ” /N (cid:1) ℓ +1 (the last formula owing to the fact that the degree of α is ℓ + 1). Hence, on combining these threeequations, we obtain α ∗ ( c g , /Nℓ ) = ( α ) ∗ ( α ) ∗ (cid:0) c g , /Nℓ (cid:1) = ( α ) ∗ h ϕ ∗ ℓ (cid:0) c g , /N (cid:1) · (cid:0) c g , “ ℓ − ” /N (cid:1) − i = (cid:16) c g , /N · ( c g , “ ℓ − ” /N ) ℓ (cid:17) · (cid:16)(cid:0) c g , “ ℓ − ” /N (cid:1) − (cid:17) ℓ +1 = c g , /N · (cid:0) c g , “ ℓ − ” /N (cid:1) − . (cid:3) Remark . In the above proposition we excluded from consideration the case when N ′ = N ℓ where ℓ | M (but ℓ ∤ N ). This case can also be treated using Kato’s methods, or deduced directly from Step 1of § op.cit. by applying the element (cid:18) −
11 0 (cid:19) , and one finds that in this case we have α ∗ (cid:0) c g , /Nℓ (cid:1) = c g , /N · ( ϕ − ℓ ) ∗ (cid:0) c g , “ ℓ − ” /N (cid:1) . However we shall solely be working with modular curves of the form Y ( m, mN ), so we will not need thisformula. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 7
Integral models of modular curves.
The following theorem is well-known:
Theorem 2.3.1 (Igusa) . There exists a smooth scheme Y ( N ) over Z [ ζ N , /N ] , representing the functorof Definition 2.1.1 on the category of Z [1 /N ] -schemes. For a sketch of the proof, see e.g. [DR73].
Proposition 2.3.2.
The Siegel units c g α,β , for all ( α, β ) ∈ ( N Z / Z ) −{ (0 , } , are elements of O ( Y ( N )) × .Proof. As shown in [Kat04, Prop 1.3], given an arbitrary scheme S , an elliptic curve E/S , and an integer c > c θ E ∈ O ( E − E [ c ]) × whose divisor is c (0) − E [ c ].As noted in [Sch98, § S is integral and E has a torsion section x : S → E of order N ,where N > c and either N is invertible on S or N has at least two prime factors, then x ∗ c θ E ∈ O ( S ) × . Applying this with S = Y ( N ), E the universal elliptic curve over S , and x the section ae + be where ( α, β ) = ( a/N, b/N ), we deduce that c g α,β extends from Y ( N ) to a unit on the integralmodel Y ( N ). (cid:3) Remark . By passage to the quotient we also see that for any b ∈ Z /N Z , b = 0, the Siegel unit c g ,b/N is a unit on the canonical Z [1 /N ]-model Y ( N ) of Y ( N ).2.4. Hecke correspondences.
We now recall how elements of the Hecke algebra can be interpretedas correspondences between modular curves, or, equivalently, as 1-cycles on a product of two modularcurves.
Lemma 2.4.1.
Let α ∈ GL +2 ( Q ) and Γ , Γ finite-index subgroups of SL ( Z ) . Then there is a uniquemorphism of varieties over Q , σ : Y (Γ ∩ α − Γ α ) → Y (Γ ) × Y (Γ ) , such that the diagram H × α ✲ H × H Y (Γ ∩ α − Γ α )( C ) ❄ σ ✲ ( Y (Γ ) × Y (Γ ))( C ) ❄ commutes (where the vertical arrows are the natural projection maps). The image of σ is an irreducibleclosed subvariety of Y (Γ ) × Y (Γ ) , and the map σ is a birational equivalence onto its image.Proof. After Definition 2.1.10 and the remarks following, the only assertion that needs checking is that σ is birational. However, by Proposition A.1.4 in the appendix (applied to the subgroups Γ and α − Γ α )we know that σ is injective away from a finite set. (cid:3) Remark . This proposition is well known in the special case Γ = SL ( Z ) and α = (cid:18) p
00 1 (cid:19) for aprime p , where it shows that Y ( p ) is the normalization of the subvariety of A cut out by the classicalmodular equation of level p ; see e.g. [DR73, § VI.6].
Lemma 2.4.3.
Let Γ , Γ ′ be as above, let α , α ∈ GL +2 ( Q ) , and for i = 1 , let C i be the curve in Y (Γ) × Y (Γ ′ ) which is the image of points of the form ( z, α i z ) . If the double cosets Γ ′ α Γ and Γ ′ α Γ aredistinct as subsets of PGL +2 ( Q ) , then C ∩ C is a finite set.Proof. Suppose P ∈ C ∩ C . Then P admits liftings to H × H of the form ( z , α z ) and ( z , α z );and since both of these points are preimages of P , we can find γ ∈ Γ and γ ′ ∈ Γ ′ such that z = γz and α z = γ ′ α z . Consequently, z is fixed by the element γ − α − γ ′ α ∈ Γ · α − α · ( α − Γ ′ α ) . By Lemma A.1.2, either γ − α − γ ′ α is the identity in PGL +2 ( Q ), in which case Γ ′ α Γ and Γ ′ α Γ havethe same projective image; or z lies in one of a finite set of orbits under the action of Γ ∩ α − Γ ′ α ,which implies that P lies in one of a finite set of points of C , as required. (cid:3) A. LEI, D. LOEFFLER, AND S.L. ZERBES
Lemma 2.4.4.
Let Γ , Γ ⊆ SL ( Z ) , and let Γ ′ ⊆ Γ and Γ ′ ⊆ Γ , with all four subgroups having finiteindex in SL ( Z ) . Let α ∈ GL +2 ( Q ) , and suppose β , . . . , β h ∈ GL +2 ( Q ) are such that we have Γ α Γ = h G i =1 Γ ′ β i Γ ′ . Let C be the curve in Y (Γ ) × Y (Γ ) which is the image of Y (Γ ∩ α − Γ α ) under the map σ ofLemma 2.4.1. Then the preimage of C in Y (Γ ′ ) × Y (Γ ′ ) is the union of h distinct curves D , . . . , D h ,where D i is the image of the map σ i : Y (Γ ′ ∩ β − i Γ ′ β i ) ✲ Y (Γ ′ ) × Y (Γ ′ ) z ✲ ( z, β i z ) . Moreover, if for each i we choose some γ i ∈ Γ such that β i ∈ Γ αγ i , then we have a commutativediagram (2) Y (Γ ′ ∩ β − i Γ ′ β i ) σ i ✲ Y (Γ ′ ) × Y (Γ ′ ) Y (Γ ∩ α − i Γ α i ) z γ i z ❄ σ ✲ Y (Γ ) × Y (Γ ) ❄ where the right-hand vertical arrow is the natural projection map.Proof. The definition of γ i implies that diagram (2) commutes, from which it is clear that D i is a liftingof C . By lemma 2.4.3, the D i are distinct.It remains only to check that the union of the D i exhausts the preimage of C . Let P ∈ C , and let˜ P be any lifting of P to H × H . Then we have P = ( γ z, γ αz ) for some γ ∈ Γ and γ ∈ Γ ; so P = ( w, γ αγ − w ), where w = γ z ∈ H . We have γ αγ − ∈ Γ ′ β i Γ ′ for some i ∈ { , . . . , h } , so inparticular the image of ˜ P in Y (Γ ′ ) × Y (Γ ′ ) lies in D i as required. (cid:3) Lemma 2.4.5.
Let Γ be a finite-index subgroup of SL ( Z ) and let Γ , Γ be finite-index subgroups of Γ such that Γ Γ = Γ . Then, in the diagram of modular curves Y (Γ ∩ Γ ) α ✲ Y (Γ ) Y (Γ ) β ❄ δ ✲ Y (Γ) γ ❄ where α, β, γ, δ are the natural projection maps, the two maps O ( Y (Γ )) × → O ( Y (Γ )) × given by β ∗ ◦ α ∗ and δ ∗ ◦ γ ∗ coincide, and similarly the maps O ( Y (Γ )) × → O ( Y (Γ )) × given by α ∗ ◦ β ∗ and γ ∗ ◦ δ ∗ coincide.Proof. Note that the hypotheses are symmetric in Γ and Γ , so it suffices to show that β ∗ ◦ α ∗ = δ ∗ ◦ γ ∗ .Moreover, since all of the morphisms in the diagram are surjective, the corresponding pullback morphismsare injective, so it suffices to show that β ∗ ◦ β ∗ ◦ α ∗ = β ∗ ◦ δ ∗ ◦ γ ∗ . Since the diagram commutes, this is equivalent to( β ∗ ◦ β ∗ ) ◦ α ∗ = α ∗ ◦ ( γ ∗ ◦ γ ∗ ) . However, the map ( β ∗ ◦ β ∗ ) is given by the product over translates by coset representatives for (Γ ∩ Γ ) \ Γ ,and the map ( γ ∗ ◦ γ ∗ ) is given by the product over coset representatives for Γ \ Γ. However, sinceΓ Γ = Γ, the natural map (Γ ∩ Γ ) \ Γ → Γ \ Γis surjective. Thus these two quotients admit a common set of coset representatives, so the two mapscoincide. (cid:3)
ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 9
Remark . One can interpret this more “categorically” as follows: our hypotheses imply that thediagram in the statement of the lemma is Cartesian (in the category of curves and dominant rationalmaps), so Y (Γ ∩ Γ ) is birational to the fibre product of Y (Γ ) and Y (Γ ) over Y (Γ). The symmetryof pushforward and pullback is then a general property of fibre products.2.5. Motivic cohomology, higher Chow groups and the Gersten complex.
We now recall thedefinition of the higher Chow group CH ( X,
1) of a variety X , and how it may be explicitly calculatedusing the Gersten complex. In this section k may be any field of characteristic 0. Let Var( k ) be thecategory of varieties over k , by which we mean separated schemes of finite type over k . Let Sm( k ) bethe full subcategory of smooth varieties. Let A = Q or Z be the coefficient ring. Definition 2.5.1 (Voevodsky, cf. [MVW06, Definition 3.4]) . Let X ∈ Sm( k ) , and p, q ∈ Z with q ≥ .Define the motivic cohomology of X to be H p M ( X, A ( q )) = H p Zar ( X, Z ( q ) ⊗ A ) , where Z ( q ) denotes Voevodsky’s motivic complex of sheaves on X , and H p Zar denotes hypercohomology(with respect to the Zariski topology).Remark . (1) We use a slightly different notation than Voevodsky; the notation used in op.cit. is H i,j ( X, A ).Our choice of notation follows [Hub00] and [Lev04].(2) Note that H p M ( X, A ( q )) is zero for p > inf(2 q, q + dim X ). It is not known to be zero for p < Theorem 2.5.3.
For any X ∈ Sm( k ) and any p, q ≥ , there is a natural isomorphsim H p ( X, Z ( q )) ∼ = CH q ( X, q − p ) . Here, the higher Chow groups are those defined by Bloch.Proof.
See [Voe02, Corollary 2] or [Lev04, Theorem 1.2]. (cid:3)
We also have an alternative description of these groups in terms of Quillen K -theory. We will actuallybe interested in the special case when p = 3 and q = 2. Here, we use a result of Landsburg [Lan91]. For X smooth over a field, m ≥ ≤ p ≤ m , he constructs a mapΨ m,p : CH m ( X, m − p ) ✲ H p ( X, K m ) , where K m is the sheafification of U K m ( U ) on X . Here, K m denotes the m -th Quillen K -group. Theorem 2.5.4.
The map Ψ m,p is an isomorphism for p = m − .Proof. See [Lan91, Theorem 2.5]. (cid:3)
Remark . For p < m − m,p may not be an isomorphism in general. As pointed outto us by Landsburg in a discussion on http://mathoverflow.net/ , if X = Spec( k ), then CH m ( X, m )is the Milnor K -group K Mm ( k ) (by a theorem of Nesterenko–Suslin) and the map Ψ m, : K Mm ( k ) → H ( X, K m ) = K m ( k ) is the natural map from Milnor to Quillen K -theory, which is not generally anisomorphism for m > Proposition 2.5.6.
Suppose that X is a smooth variety of finite type over a field k . Then there is aresolution of the sheaf K m ✲ K m ✲ a x ∈ X ( i x ) ∗ K m ( k ( x )) ✲ a x ∈ X ( i x ) ∗ K m − ( k ( x )) ✲ . . . Proof.
See [Qui73]. (cid:3)
Corollary 2.5.7.
The group H ( X, K ) is the first homology group of the “Gersten complex” (3) Gerst ( X ) : a x ∈ X K ( k ( x )) d ✲ a x ∈ X k ( x ) × d ✲ a x ∈ X Z , where d is the tame symbol map, and d maps a function to its divisor (c.f. [Fla92, Section 2] ). Combining the above results, we get the following statement.
Proposition 2.5.8.
Assume that X is a smooth variety of finite type over a field k . Then we haveisomorphisms H (Gerst ( X )) ∼ = H ( X, K ) ∼ = CH ( X, ∼ = H M ( X, Z (2)) . We shall use these to identify CH ( X,
1) with H (Gerst ( X )); it is the latter group in which we shallactually construct elements. Notation.
We shall write Z ( X,
1) to denote the kernel of the boundary map d in the Gersten complexGerst ( X ), so Z ( X,
1) = (X i ( C i , φ i ) : C i ∈ X , φ i ∈ k ( C i ) × , X i div( φ i ) = 0 ) . This is a slight abuse of notation, since in Bloch’s theory of higher Chow groups Z ( X,
1) is used todenote something slightly different (a certain subgroup of the codimension 2 cycles on X × A ); but weshall not use Bloch’s construction directly in this paper, so this abuse should cause no confusion. Remark . We shall, in fact, construct an “Euler system” in the groups Z ( X,
1) as X varies over afamily of modular surfaces; that is, our compatibility properties will hold at the level of cycles, ratherthan just after quotienting out by the image of tame symbols. The groups Z ( X,
1) are much easierto work with, as they have good descent properties: for a finite surjective map X → Y , the pullback Z ( Y, → Z ( X,
1) is injective.This is, in a sense, analogous to the fact that in the construction of [Kat04] the compatibility propertiesof the Euler system in K of modular curves are proved at the level of K ⊗ K , before quotienting byelements of the form x ⊗ (1 − x ).2.6. Zeta elements on Y ( m, mN ) . We begin by defining elements of Z ( Y ( m, mN ) , zeta elements . Definition 2.6.1.
For m, N ≥ , the curve C m,N,j ⊆ Y ( m, mN ) is defined as the subvariety (cid:18) u, v : v = (cid:18) j (cid:19) u (cid:19) . For c > coprime to mN , we define c Z m,N,j = ( C m,N,j , φ ) ∈ Z ( Y ( m, mN ) , , where φ ∈ O ( C ) × is the pullback of c g , /mN along either of the projections C m,N,j → Y ( m, mN ) . The first properties of these elements are the following.
Proposition 2.6.2.
The elements c Z m,N,j have the following properties:(1) We have ρ ∗ c Z m,N,j = c Z m,N, − j , where ρ is the involution of Y ( m, mN ) which interchanges thefactors.(2) For c, d > coprime to mN , the element h d − (cid:0)(cid:0) d d (cid:1) , (cid:0) d d (cid:1)(cid:1) ∗ i · c Z m,N,j is symmetric in c and d . In particular, there exists a unique element Z m,N,j ∈ Z ( Y ( m, mN ) , ⊗ Q such that c Z m,N,j = (cid:2) c − (( c c ) , ( c c )) ∗ (cid:3) Z m,N,j for any c .(3) We have (cid:18)(cid:18) b
00 1 (cid:19) , (cid:18) b
00 1 (cid:19)(cid:19) ∗ c Z m,N,j = c Z m,N,b − j for any b ∈ ( Z /mN Z ) × . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 11
Proof.
Part (1) is obvious, and part (2) follows immediately from the fact that the Siegel units c g α,β satisfy ( d c g α,β − c g dα,dβ ) = ( c d g α,β − d g cα,cβ )for any α, β ∈ N Z / Z − { (0 , } and any c, d > mN (cf. [Kat04, Proposition 1.3(2)]). Wemay then define Z m,N,j = ( c − − c Z m,N,j for any c > mN .Property (3) follows from the identity (cid:18) b
00 1 (cid:19) − (cid:18) j (cid:19) = (cid:18) b − j (cid:19) (cid:18) b
00 1 (cid:19) − . (cid:3) Generalized Beilinson–Flach elements.
The Beilinson–Flach elements of [BDR12] are elementsof CH ( Y ( N ) ,
1) defined as (∆ , φ ), where ∆ is the diagonal and φ is a suitable modular unit on ∆. Ourgeneralization of this is motivated by the observation that one can recover the twists of a modular formby Dirichlet characters modulo m from the “shifted” forms f ( x + a/m ) for a ∈ ( Z /m Z ) × ; this is alsothe idea underlying the construction of the p -adic L -function of a single modular form using modularsymbols. Lemma 2.7.1.
Let m, N ≥ with m N ≥ , and j ∈ Z . Then there is a unique morphism of algebraicvarieties over C , κ j : Y ( m N ) ⊗ C → Y ( N ) ⊗ C , such that the diagram of morphisms of complex-analytic manifolds H z z + j/m ✲ H Y ( m N )( C ) ❄ κ j ✲ Y ( N )( C ) ❄ commutes. The morphism κ j is defined over Q ( µ m ) , and depends only on the residue class of j mod m .Proof. The existence of such a map at the level of quotients of H follows immediately from the inclusionof matrix groups (cid:18) jm (cid:19) Γ ( m N ) (cid:18) − jm (cid:19) ⊆ Γ ( N ) . However, in order to descend to an algebraic morphism over Q ( µ m ) we use the canonical models above.We first consider the map Y ( m N ) → Y ( m, mN ) which maps ( E, e , e ) to ( E/ h me i , [ mN e ] , [ e ]).This factors through the quotient by the subgroup (cid:18) u ∗ (cid:19) : u = 1 mod m Z , which we have identifiedwith Y ( m N ) ⊗ Q ( µ m ). This map is compatible with z mz on H . We now consider the composition Y ( m N ) × µ m ✲ Y ( m, mN ) (cid:16) j (cid:17) ✲ Y ( m, mN ) t m ✲ Y ( N ) × µ m , where t m is as in Definition 2.1.6. All three morphisms are maps of Q ( µ m )-varieties (i.e. they commutewith the projection maps to µ m ); and on the fibre over ζ m ∈ µ m ( C ) they correspond to z mz , z z + j , and z z/m , so the composition corresponds to z z + j/m . (cid:3) Definition 2.7.2.
For m, N, j as above, let ι m,N,j be the map (1 , κ j ) : Y ( m N ) × µ m → Y ( N ) × µ m , and C m,N,j the irreducible curve in Y ( N ) that is the image of ι m,N,j . We shall now use these curves C m,N,j to define a class in CH ( Y ( N ) × µ m , Definition 2.7.3.
Let N ≥ , m ≥ , j ∈ Z /m Z as above. Let c ≥ be coprime to mN and let α ∈ Z /m N Z . We define the generalized Beilinson–Flach element c Ξ m,N,j,α ∈ CH ( Y ( N ) ⊗ Q ( µ m ) , as the class of the pair (cid:16) C m,N,j , ( ι m,N,j ) ∗ ( c g ,α/m N ) (cid:17) ∈ Z ( Y ( N ) × µ m , . When α = 1 we drop it from the notation and write simply c Ξ m,N,j . The following proposition shows that these zeta elements are simply the “ Y -versions” of those definedin the previous section. Proposition 2.7.4.
The generalized Beilinson–Flach element c Ξ m,N,j,α is the pushforward of the element (cid:18)(cid:18) α α (cid:19) , (cid:18) α α (cid:19)(cid:19) ∗ c Z m,N,j ∈ Z ( Y ( m, mN ) , along the map t m × t m : Y ( m, mN ) → Y ( N ) × µ m introduced in § It is clear from the construction of the map κ m,N,j that C m,N,j is the image of C m,N,j under t m × t m . So it suffices to show that the pushforward of c g , /m N from Y ( m N ) ⊗ Q ( µ m ) to Y ( m, mN )along the map constructed above is c g , /mN .Let U be the subgroup of GL ( Z /m N Z ) consisting of elements (cid:18) a bc d (cid:19) which satisfy c = 0 , d =1 mod m N and a = 1 mod m (and b arbitrary). This is clearly contained in the subgroup U ′ of elementssatisfying a = 1 mod m , c = 0 mod m N and d = 1 mod mN , and a set of coset representatives for U/U ′ is given by the matrices (cid:26)(cid:18) mN t (cid:19) : 0 ≤ t < m (cid:27) . Hence the pushforward of c g , /m N from U \ Y ( m N ) to U ′ \ Y ( m N ) is given by Y ≤ t 00 1 (cid:19) sends U ′ tothe subgroup U ′′ = (cid:26)(cid:18) a bc d (cid:19) : a = 1 , b = 0 mod m,c = 0 , d = 1 mod mN (cid:27) , and we have U ′′ \ Y ( m N ) = Y ( m, mN ). (cid:3) We now record some properties of the generalized Beilinson–Flach elements. Proposition 2.7.5. The elements above have the following properties:(1) The element c Ξ m,N,j,α only depends on the congruence class of α modulo mN (not m N ).(2) The involution of Y ( N ) ⊗ Q ( µ m ) given by switching the two factors interchanges c Ξ m,N,j and c Ξ m,N, − j .(3) For q ∈ ( Z /m Z ) × , we have σ ∗ q ( c Ξ m,N,j,α ) = c Ξ m,N,q − j,α , where σ q ∈ Gal( Q ( µ m ) / Q ) is thearithmetic Frobenius at q .(4) For any r ∈ ( Z /mN Z ) × , we have c Ξ m,N,j,rα = h d × d i ∗ c Ξ m,N,k,α where k = r − j ∈ Z /m Z , d is the image of r in ( Z /N Z ) × , and h d × d i denotes the action on Y ( N ) ⊗ Q ( µ m ) of the element (cid:18) d − d (cid:19) × (cid:18) d − d (cid:19) ∈ SL ( Z /N Z ) . (5) For c, d coprime to mN , the expression d c Ξ m,N,j,α − c Ξ m,N,j,dα is symmetric in c and d . In particular, there exist well-defined elements Ξ m,N,j,α ∈ CH ( Y ( N ) ⊗ Q ( µ m ) , ⊗ Q such that we have c Ξ m,N,j,α = c Ξ m,N,j,α − Ξ m,N,j,cα = ( c − h c × c i ∗ σ c ) Ξ m,N,j,α . Proof. After Proposition 2.7.4, parts (1) and (2) are immediate. The remaining statements follow fromProposition 2.6.2, together with the fact that t m intertwines the action of ( d 00 1 ) on Y ( m, mN ) with thearithmetic Frobenius σ d on Y ( N ) × µ m (Proposition 2.1.7(2)). (cid:3) ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 13 Cuspidal components. In the preceding sections we have constructed elements of the higherChow groups of affine surfaces. In order to be able to apply results on regulator maps, it is convenientto have elements of Chow groups of projective surfaces instead. We shall show that this can be achieved,but not in a canonical way, and only at the cost of tensoring with Q . Theorem 2.8.1. Let N, m, j be as in Definition 2.7.3. Then the element Ξ m,N,j of CH ( Y ( N ) ⊗ Q ( µ m ) , ⊗ Q is in the image of the pullback map CH ( X ( N ) ⊗ Q ( µ m ) , ⊗ Q → CH ( Y ( N ) ⊗ Q ( µ m ) , ⊗ Q induced by the open embedding Y ( N ) ֒ → X ( N ) . We will actually prove a slightly more precise statement, see Proposition 2.8.5 below.Recall that we constructed c Ξ m,N,j as the class in CH ( Y ( N ) ⊗ Q ( µ m ) , 1) of an explicit elementof Z ( Y ( N ) ⊗ Q ( µ m ) , c Y m,N,j . It is clear that we may alsoregard c Y m,N,j as an element of Gerst ( X ( N ) ⊗ Q ( µ m )), whose divisor is not necessarily trivial, butis supported on the cuspidal locus.We will need a preparatory lemma. Let K be any number field. Definition 2.8.2. We shall call an element of Gerst ( X ( N ) ⊗ K ) negligible if it is supported on afinite union of curves of the form { c } × X ( N ) or X ( N ) × { d } for points c, d ∈ X ( N ) \ Y ( N ) .Remark . Here by “point” we mean a 0-dimensional point of X ( N ) \ Y ( N ) considered as a K -scheme, i.e. a Gal( K/K )-orbit of points in the naive sense. Note that this is slightly more restrictivethan the definition of “negligible” in [BDR12].Before proving the theorem, we will need the following preparatory lemma: Lemma 2.8.4. Let K be any number field and let u, v, x, y be cuspidal points of X ( N ) ⊗ K . Then thereexists a negligible element in Gerst ( X ( N ) ⊗ K ) ⊗ Q with divisor ( u, v ) − ( x, y ) .Proof. By the Manin–Drinfeld theorem [Dri73], there exist elements f, g ∈ O ( Y ( N ) ⊗ K ) × ⊗ Q whosedivisors are v − y and u − x , respectively. The the element( { u } × X ( N ) , f ) + ( X ( N ) × { y } , g )has the required property. (cid:3) We can now prove the following proposition: Proposition 2.8.5. Let N, m, j, c be as in Definition 2.7.3. Then there exists an integer r ≥ and anegligible element Θ such that R · c Y m,N,j + Θ ∈ Z (Gerst ( X ( N ) ⊗ Q ( µ m ))) . Proof. Recall that ι m,N,j is the map ι m,N,j = (1 , κ j ) : Y ( m N ) → Y ( N ) , where κ j is induced from the map H → H given by z z + jm . It follows that, if we regard c Y m,N,j as an element of Z ( X ( N ) ⊗ Q ( µ m ) , (cid:0) c Y m,N,j (cid:1) is a linear combination of divisors ofthe form ( c , c + jm ) − ( c , c + jm ). But Lemma 2.8.4 implies that there exists a negligible elementΘ ∈ Gerst ( X ( N ) ⊗ Q ( µ m )) such that Div(Θ) = Div (cid:0) c Y m,N,j (cid:1) . Then the element c X m,N,j := c Y m,N,j − Θ ∈ Z ( X ( N ) ⊗ Q ( µ m ) , ⊗ Q has the required properties. (cid:3) This clearly implies Theorem 2.8.1. Remark . (1) Note that the negligible element Θ is not uniquely determined. However, as we will see below,this will not matter for the evaluation of the element via the Beilinson or the syntomic regulator.(2) Since X ( N ) and Y ( N ) have the same rational function field, any element of CH ( X ( N ) ⊗ Q ( µ m ) , ⊗ Q lifting c Ξ m,N,j is necessarily the class of an element of Z ( X ( N ) ⊗ Q ( µ m ) , ⊗ Q differing from c Y m,N,j by a negligible element.(3) Since there are only finitely many cusps on X ( N ), the constant R may be chosen to be inde-pendent of c , m and j , although it may of course depend on N . Zeta elements versus generalized Beilinson–Flach elements. At the referee’s request, weshall briefly clarify the relations between the two classes of elements we have introduced (the zeta elements c Z m,N,j and the generalized Beilinson–Flach elements c Ξ m,N,j ) and how they would relate to a hypo-thetical “optimal” construction. Recall that the element c Z m,N,j lies in the group CH ( Y ( m, mN ) , c Ξ m,N,j ∈ CH ( Y ( N ) ⊗ Q ( µ m ) , 1) is the pushforward of c Z m,N,j via the morphism t m × t m of § c Ξ m,N,j which will be used in §§ c Z m,N,j is that they are somewhat easier to work with thanthe c Ξ m,N,j . In the next section we shall prove norm-compatibility relations for the c Z m,N,j , and deducenorm relations for the c Ξ m,N,j as a consequence; given the somewhat opaque map ι m,N,j entering into thedefinition of the elements c Ξ m,N,j , it seems unlikely that these norm relations could be proved withoutthe introduction of some auxilliary higher-level modular curve.A second reason to consider the elements c Z m,N,j is the following optimistic idea. Let us fix a prime p , and a level N coprime to p , and consider the curves Y ( p r , N p r ) for r ≥ 0, and their self-products Y ( p r , N p r ) . These form a tower of surfaces with Galois group GL ( Z p ) × det GL ( Z p ). Let us imaginethat we could construct a norm-compatible family of elements in the higher Chow groups of this tower,analogous to the compatible family of elements in K of the GL ( Z p )-tower of modular curves constructedby Kato in [Kat04]. Then one could potentially perform a “nonabelian twisting” operation analogousto equation (8.4.3) of op.cit. in order to obtain classes in the cohomology groups attached to pairs ofmodular forms of arbitrary weights k, ℓ ≥ c Z p r ,N,j represent our best attempt to realize this dream. They do indeed live onthe surfaces Y ( p r , N p r ) ; but the norm-compatibility relation they satisfy (Theorem 3.3.1) involves the“twisted” degeneracy map τ p : Y ( p r +1 , N p r +1 ) → Y ( p r , N p r ) of Definition 2.1.6, given by z z/p on the upper half-plane H , rather than the natural one corresponding to the identity map on H . Thenorm-compatibility relation also involves a Hecke operator at p , which does not appear in the settingof [Kat04]. Consequently, our methods will only allow us to construct cohomology classes for Rankin–Selberg convolutions of higher weight forms under additional ordinarity assumptions, when we can useHida’s theory of p -adic families in order to pass from weight 2 to general weights.3. Norm relations for generalized Beilinson–Flach elements The first norm relation: varying N . We now consider the relation between the zeta elementsat different levels N (for fixed m and j ). Theorem 3.1.1 (First norm relation) . Let α be the natural projection Y ( m, mN ′ ) → Y ( m, mN ) , where N and N ′ are positive integers such that N | N ′ .(1) If prime( N ′ ) ⊆ prime( mN ) , the pushforward map ( α × α ) ∗ : CH ( Y ( m, mN ′ ) , → CH ( Y ( m, mN ) , maps c Z m,N ′ ,j to c Z m,N,j .(2) If N ′ = N ℓ , where ℓ ∤ mN is prime, then ( α × α ) ∗ ( c Z m,Nℓ,j ) = h − (cid:16)(cid:16) ℓ − ℓ − (cid:17) , (cid:16) ℓ − ℓ − (cid:17)(cid:17) ∗ i c Z m,N,j , where (cid:16) ℓ − ℓ − (cid:17) is considered as an element of GL ( Z /mN Z ) .Proof. It is clear that the map α commutes with the action of (cid:0) j (cid:1) , so we have ( α × α )( C m,N ′ ,j ) = C m,N,j ;more precisely, we have a commutative diagram Y ( m, mN ′ ) (1 , (cid:0) j (cid:1) ) ✲ C m,N ′ ,j Y ( m, mN ) α ❄ (1 , (cid:0) j (cid:1) ) ✲ C m,N,j .α × α ❄ ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 15 From Theorem 2.2.4, we know that if prime( mN ′ ) = prime( mN ), then α ∗ (cid:0) c g , /mN ′ (cid:1) = c g , /mN , sopart (1) of the theorem follows. For part (2), we deduce from the second part of Theorem 2.2.4 that( α × α ) ∗ ( c Z m,N ′ ,j ) = c Z m,N,j − c Z m,N,j, “ ℓ − ” , where we write c Z m,N,j,a for the element formed with c g ,a/mN in place of c g , /mN . However, we have c g , “ ℓ − ” /mN = (cid:16) ℓ − ℓ − (cid:17) ∗ c g , /mN as elements of O ( Y ( m, mN )) × , and the action of (cid:16) ℓ − ℓ − (cid:17) evidently commutes with that of (cid:0) j (cid:1) . (cid:3) We now deduce a compatibility relation for zeta elements on Y ( N ) ⊗ Q ( µ m ). Theorem 3.1.2 (First norm relation on Y ( N )) . Let α be the natural projection Y ( N ′ ) → Y ( N ) , where N, N ′ are positive integers such that N | N ′ .If prime( mN ′ ) = prime( mN ) then we have ( α × α ) ∗ ( c Ξ m,N ′ ,j ) = c Ξ m,N,j . If N ′ = ℓN where ℓ ∤ mN , then we have ( α × α ) ∗ ( c Ξ m,N ′ ,j ) = (cid:2) − ( h ℓ − i , h ℓ − i ) ∗ σ − ℓ (cid:3) c Ξ m,N,j , where σ ℓ denotes the arithmetic Frobenius at ℓ .Proof. This follows immediately from Theorem 3.1.1, since the map π m,N : Y ( m, mN ) → Y ( N ) × µ m intertwines (cid:0) ℓ − ℓ (cid:1) with the diamond operator h ℓ i , and ( ℓ 00 1 ) with the Frobenius σ ℓ . (cid:3) Hecke operators. We define Hecke operators, following [Kat04, §§ ℓ be prime, and M, N ≥ ℓ | M or ℓ | N ). We define a correspondence on Y ( M, N ) as follows. We have adiagram of modular curves Y ( M ( ℓ ) , N ) Y ( M, N ) π ❄ Y ( M, N ) , π ✲ where π is the natural degeneracy map, corresponding to the identity on H , and π is the “twisted”degeneracy map, corresponding to z z/ℓ on H . (In the notation introduced in the proof of Theorem2.2.4 above, π was denoted pr , and π is the composite of ϕ − ℓ : Y ( M ( ℓ ) , N ) → Y ( M, N ( ℓ )) with thenatural projection Y ( M, N ( ℓ )) → Y ( M, N )).We denote the correspondence ( π ) ∗ ( π ) ∗ by T ′ ℓ if ℓ ∤ M N , and by U ′ ℓ if ℓ | M N . We denote theoperator ( π ) ∗ ( π ) ∗ by T ℓ (resp. U ℓ ); these latter operators T ℓ , U ℓ are the familiar Hecke operators ofthe transcendental theory, but it is the T ′ ℓ , U ′ ℓ that will concern us most here.3.3. The second norm relation for ℓ | N . Our goal in this section is to prove the following theorem: Theorem 3.3.1 (Second norm relation, ℓ | N case) . Let m ≥ , N ≥ , and ℓ a prime dividing N . Let τ ℓ denote the degeneracy map Y ( mℓ, mℓN ) → Y ( m, mN ) of Definition 2.1.6, compatible with z z/ℓ on H . Then for any j ∈ ( Z /ℓm Z ) × , and c > coprime to ℓmN , we have ( τ ℓ × τ ℓ ) ∗ ( c Z ℓm,N,j ) = ( ( U ′ ℓ × U ′ ℓ ) ( c Z m,N,j ) if ℓ | m , ( U ′ ℓ × U ′ ℓ − ∆ ∗ ℓ ) ( c Z m,N,j ) if ℓ ∤ m .where ∆ ℓ denotes the action of any element of GL ( Z /mN Z ) of the form (cid:18)(cid:18) x 00 1 (cid:19) , (cid:18) x 00 1 (cid:19)(cid:19) with x = ℓ mod m . We shall prove Theorem 3.3.1 below. First, we note that it implies the following property of thegeneralized Beilinson–Flach elements c Ξ m,N,j on Y ( N ): Theorem 3.3.2 (Second norm relation on Y ( N ), ℓ | N case) . Let m ≥ , N ≥ , ℓ a prime dividing N , j ∈ ( Z /ℓm Z ) × , and c ∈ ( Z /ℓmN Z ) × . Then we have norm ℓmm ( c Ξ ℓm,N,j ) = ( ( U ′ ℓ × U ′ ℓ ) ( c Ξ m,N,j ) if ℓ | m , ( U ′ ℓ × U ′ ℓ − σ ℓ ) ( c Ξ m,N,j ) if ℓ ∤ m ,where norm ℓmm denotes the Galois norm map, and σ ℓ , for ℓ ∤ m , denotes the arithmetic Frobenius at ℓ in Gal( Q ( µ m ) / Q ) .Proof of Theorem 3.3.2 (assuming Theorem 3.3.1). Let t m × t m : Y ( m, mN ) → Y ( N ) ⊗ Q ( µ m ) bethe map of § U ′ ℓ , U ′ ℓ ) on both sides, and intertwines theaction of ∆ ℓ with the arithmetic Frobenius σ ℓ . Since c Ξ m,N,j = ( t m × t m ) ∗ ( c Z m,N,j ) by Proposition2.7.4, Theorem 3.3.2 follows from Theorem 3.3.1. (cid:3) Proof of Theorem 3.3.1. Since we are assuming ℓ | N , let us write N ′ = N/ℓ . We have the followingcommutative diagram of modular curves: Y ( ℓm, ℓmN ) α ✲ Y ( ℓm, mN ) pr ✲ Y ( m ( ℓ ) , mN ) Y ( m, mN ) ✛ π τ ℓ ✲ Y ( m, mN ) .π ❄ Here α is the natural projection Y ( ℓm, ℓmN ) → Y ( ℓm, ℓmN ′ ) = Y ( ℓm, mN ), and pr is the naturalprojection map. Consequently, we have a commutative diagram of surfaces Y ( ℓm, ℓmN ) α × α ✲ Y ( ℓm, mN ) pr × pr ✲ Y ( m ( ℓ ) , mN ) Y ( m, mN ) ✛ π × π τ ℓ × τ ℓ ✲ Y ( m, mN ) .π × π ❄ Applying Theorem 3.1.1, we see that ( α × α ) ∗ c Z ℓm,N,j = c Z ℓm,N ′ ,j . Since Y ( m ( ℓ ) , mN ) is thequotient of Y ( ℓm, mN ) by the subgroup (cid:26)(cid:18)(cid:18) x 00 1 (cid:19) , (cid:18) x 00 1 (cid:19)(cid:19) : x ∈ Z /ℓm Z , = 1 mod m (cid:27) . Thus we have (pr × pr) ∗ (pr × pr) ∗ ( c Z ℓm,N ′ ,j ) = X x ∈ Z /ℓm Z x =1 mod m c Z ℓm,N ′ ,xj . Let us now compute (pr × pr) ∗ ( π × π ) ∗ c Z m,N,j . Since Y ( m, mN ) is the quotient of Y ( ℓm, mN ) by the group (cid:26)(cid:18)(cid:18) x y (cid:19) , (cid:18) x z (cid:19)(cid:19) : x, y, z ∈ Z /ℓm Z , x = 1 mod m,y, z = 0 mod m (cid:27) , we see that the preimage of C m,N,k is the union of the curves C ℓm,N ′ ,k , for k ∈ Z /ℓm Z congruent to j modulo m , each of which is isomorphic to Y ( ℓm, mN ). By counting degrees, they must be distinct. Themodular units c g , /mN and c g , /ℓmN ′ coincide, and thus we have(pr × pr) ∗ ( π × π ) ∗ c Z m,N,j = X k ∈ Z /ℓm Z k = j mod m c Z ℓm,N ′ ,k . By hypothesis, j is invertible modulo ℓm . Thus if ℓ | m , the sets { xj : x = 1 mod m } and { k : k = j mod m } coincide, and since (pr × pr) ∗ is clearly injective, we conclude that(pr × pr) ∗ ( c Z ℓm,N ′ ,j ) = ( π × π ) ∗ c Z m,N,j . Applying ( π × π ) ∗ gives the result in this case. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 17 If ℓ ∤ m , there is exactly one lifting j of j to Z /ℓm Z which is not a unit. The matrix (cid:18) j (cid:19) normalizes the subgroup of GL ( Z /mN Z ) corresponding to Y ( m ( ℓ ) , mN ), and thus defines a curve A in Y ( m ( ℓ ) , mN ) which is isomorphic to Y ( m ( ℓ ) , mN ), consisting of points ( u, v ) with v = (cid:18) j (cid:19) u ;and we have ( π × π ) ∗ c Z m,N,j = (pr × pr) ∗ ( c Z ℓm,N ′ ,j ) + ( A , c g , /mN ) . The image of A under π is C m,N,ℓ − j ; moreover, we have a diagram Y ( m ( ℓ ) , mN ) ∼ = ✲ A Y ( m, mN ) π ❄ ∼ = ✲ C m,N,ℓ − j .π × π ❄ We claim that ( π ) ∗ (cid:0) c g , /mN (cid:1) = c g , /mN . However, ( π ) ∗ c g , /mN is the pushforward of ϕ ∗ ℓ ( c g , /mN ) ∈O ( Y ( m, mN ( ℓ ))) × along the natural projection O ( Y ( m, mN ( ℓ ))) × → O ( Y ( m, mN )) × , and the distri-bution relation of Equation (1b) shows that the pushforward of ϕ ∗ ℓ ( c g , /mN ) is c g , /mN , as required.Hence ∈ Z ( Y ( m ( ℓ ) , mN ) , 1) is( π × π ) ∗ ( A , c g , /mN ) = ( C m,N,ℓ − j , c g , /mN ) = c Z m,N,ℓ − j = ∆ ∗ ℓ ( c Z m,N,j ) , as required. (cid:3) The second norm relation for p ∤ mN . In this section, we shall assume that N ≥ m ≥ j ∈ Z /m Z , and p is a prime such that p ∤ mN . Our aim is to prove the following theorem: Theorem 3.4.1. We have X k ∈ Z /mp Z k = j mod mp ∤ k c Ξ mp,N,k = (cid:16) − σ p + ( T ′ p , T ′ p ) + (cid:2) ( p + 1)( h p − i , h p − i ) − ( h p − i , T ′ p ) − ( T ′ p , h p − i ) (cid:3) σ − p + (cid:0) h p − i T ′ p , h p − i T ′ p (cid:1) σ − p − p (cid:0) h p − i , h p − i (cid:1) σ − p (cid:17) c Ξ m,N,j . Remark . One can formulate a version of this theorem for the zeta elements c Z m,N,j , from whichTheorem 3.4.1 would follow in the same way as Theorem 3.3.2 follows from Theorem 3.3.1. The argumentgiven below can easily be extended to prove this slightly stronger result; however, we shall not pursuethis here, as the above statement suffices for our applications.We begin the proof of Theorem 3.4.1 by rewriting the T ′ p terms using a related Hecke operator S ′ p . Proposition 3.4.3. As elements of the Hecke algebra of Γ ( N ) , we have T ′ p = S ′ p + ( p + 1) h p − i R p ,where S ′ p is the double coset of (cid:18) p 00 1 (cid:19) and R p is the double coset of (cid:18) p p (cid:19) .Proof. This is a simple computation from the definition of multiplication in the Hecke algebra. (cid:3) Since R p acts trivially on everything in sight, the formula of Theorem 3.4.1 can be written as(4) X k ∈ ( Z /mp Z ) × k = j mod m c Ξ mp,N,k = (cid:16) (cid:0) ( T ′ p , T ′ p ) − σ p − p ( h p − i , h p − i ) σ − p (cid:1) (cid:0) (cid:0) h p − i , h p − i (cid:1) σ − p (cid:1) − (cid:2) ( h p − i , S ′ p ) + ( S ′ p , h p − i ) (cid:3) σ − p (cid:17) c Ξ m,N,j . Evaluation of ( T ′ p , T ′ p ) c Ξ m,N,j . First we shall make a careful study of the operator ( T ′ p , T ′ p ). Proposition 3.4.4. If G = SL ( Z /p Z ) and B is the lower-triangular Borel subgroup, then B \ G/B hasexactly 2 elements, B and its complement (the “big Bruhat cell”).Proof. Well-known. (cid:3) Corollary 3.4.5. Let Γ be any congruence subgroup of SL ( Z ) of level prime to p , and let α ∈ SL ( Q ) be integral at p . Then the double coset Γ α Γ is the union of exactly two double cosets of Γ ′ = Γ ∩ Γ ( p ) , corresponding to those elements whose reductions modulo p land in the two double cosets of B in SL ( Z /p Z ) .Proof. This is a consequence of strong approximation for SL ( Z ). Since Γ has level prime to p , it surjectsonto SL ( Z /p Z ). Hence we may assume (by left or right multiplying α by an appropriate element of Γ)that the reduction of α modulo p is the identity.We first note that Γ ∩ α − Γ α is also a congruence subgroup of level prime to p , so (by the strongapproximation theorem) we see that Γ \ Γ ′ admits a set of coset representatives lying in Γ ∩ α − Γ α andthus Γ α Γ = Γ α Γ ′ .Now let x = γαγ ′ ∈ Γ α Γ ′ . We consider the reduction ¯ x of x modulo p . If this lies in B , then (since ¯ α and ¯ γ ′ are in B ) we must have ¯ x ∈ B and hence γ ∈ Γ ′ ; thus x ∈ Γ ′ α Γ ′ .On the other hand, let µ be any element of Γ which is not in Γ ( p ). If ¯ γ / ∈ B , then ¯ γ ∈ B ¯ µB ; sothere is some σ ∈ Γ ∩ α − Γ α ∩ Γ ( p ) such that ¯ γ ∈ B ¯ µ ¯ σ . So ¯ γ ¯ σ − ¯ µ − ∈ B and thus γ ∈ Γ ′ µσ . Hence x ∈ Γ ′ µσα Γ ′ ; but α − σα ∈ Γ ′ (since by hypothesis α = 1 mod p and thus conjugation by α fixes Γ ( p ))and thus x ∈ Γ ′ µα Γ ′ . (cid:3) Corollary 3.4.6. For k ∈ Z /mp Z , let D m,N,k denote the curve in Y (Γ ( N ) ∩ Γ ( p )) consisting ofpoints of the form (cid:0) z, z + km (cid:1) . Then the preimage π − ( C m,N,j ) ⊆ Y (Γ ( N ) ∩ Γ ( p )) consists of exactly2 components: one is the curve D m,N,k where k is the unique lifting of j to Z /mp Z which is zero mod p , and the other is the curve D m,N,k where k is any lifting of j to Z /mp Z which is a unit modulo p (theresulting curve being independent of the choice of lifting).Proof. The preimage of C m,N,j in H × H is exactly the set of ( u, v ) such that v = γu for some γ in thedouble coset Γ ( N ) (cid:18) j/m (cid:19) Γ ( N ). The above proposition describes the decomposition of this setinto double cosets of Γ ( N ) ∩ Γ ( p ), hence the result. (cid:3) For any k lifting j (unit or non-unit), we may erect the following diagram of modular curves: Y (Γ ( mpN ) ∩ Γ ( mp )) Y (Γ ( mN ) ∩ Γ ( m ) ∩ U k ) ✛ ρ k Y (Γ ( mN ) ∩ Γ ( m )) ✛ α D m,N,k λ k ✲ C m,N,j ✛ π ι ′ m , N , j ✲ C mp,N,k ι ′ m p , N , k ✲ π ✲ Here U k is the preimage in SL ( Z ) of the subgroup B ∩ (cid:18) k m (cid:19) − B (cid:18) k m (cid:19) = B ∩ (cid:18) k (cid:19) − B (cid:18) k (cid:19) . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 19 (The equality follows from the fact that conjugation by (cid:18) m (cid:19) fixes B .) This subgroup is just B if k ∈ p Z ; otherwise it is the subgroup (cid:26)(cid:18) u − k − ( u − u − ) u (cid:19) : u ∈ ( Z /p Z ) × (cid:27) , which is a maximal torus in SL ( Z /p Z ). The square at the bottom left of the diagram is Cartesian. Themaps α , ρ k and π are the natural projection maps, and the remaining maps are defined by ι ′ m,N,j : z (cid:0) zm , z + jm (cid:1) ι ′ mp,N,k : z (cid:16) zmp , z + kmp (cid:17) π : ( u, v ) (cid:16) up , vp (cid:17) λ k : z (cid:0) zm , z + km (cid:1) Definition 3.4.7. Let a and b be the unique elements of Z /pm Z congruent to j modulo m and such that a = 0 mod p and b = 1 mod p . An application of Lemma 2.4.5 shows that we have: Corollary 3.4.8. For any α ∈ Z /mN Z , we have ( T ′ p , T ′ p )( c Ξ m,N,j,α ) = (cid:0) C mp,N,a , ( π ◦ λ a ) ∗ c g ,α/mN (cid:1) + (cid:0) C mp,N,b , ( π ◦ λ b ) ∗ c g ,α/mN (cid:1) . (It is convenient to allow α = 1 here, for reasons that will become clear below.)We first consider the term for a . Here we have U a = Γ ( p ), so Γ ( mN ) ∩ Γ ( m ) ∩ U a = Γ ( mN ) ∩ Γ ( mp ). Since p | a , we see that π ◦ λ a can also be expressed as a composition(5) Y (Γ ( mN ) ∩ Γ ( mp )) Y (Γ ( mN ) ∩ Γ ( m )) z z/p ❄ C mp,N,a = C m,N, “ p − ” j z (cid:16) zm , z +“ p − ” jm (cid:17) ❄ where “ p − ” is the inverse of p in Z /m Z . Proposition 3.4.9. The pushforward of c g ,α/mN to Y (Γ ( mN ) ∩ Γ ( m )) along the first map in (5) is c g ,α/mN · (cid:16) c g , “ p − ” α/mN (cid:17) p .Proof. See [Kat04, 2.13.2]. (cid:3) The second map in (5) is just ι ′ m,N, “ p − ” j , so we deduce that (cid:0) C mp,N,a , ( π ◦ λ a ) ∗ c g ,α/mN (cid:1) = (cid:16) C m,N, “ p − ” a , ( ι ′ m,N, “ p − ” j ) ∗ (cid:16) c g ,α/mN · (cid:16) c g , “ p − ” α/mN (cid:17) p (cid:17) (cid:17) = c Ξ m,N, “ p − ” j,α + p c Ξ m,N, “ p − ” j, “ p − ” α = ( σ p + p ( h p − i , h p − i ) σ − p ) c Ξ m,N,j,α . Corollary 3.4.10. For any α ∈ ( Z /mN Z ) × , we have (cid:0) ( T ′ p , T ′ p ) − σ p − p ( h p − i , h p − i ) σ − p (cid:1) c Ξ m,N,j,α = (cid:16) C mp,N,b , ( π ◦ λ b ) ∗ (cid:0) c g ,α/mN (cid:1) (cid:17) . In particular, (cid:0) ( T ′ p , T ′ p ) − σ p − p ( h p − i , h p − i ) σ − p (cid:1) (cid:0) (cid:0) h p − i , h p − i (cid:1) σ − p (cid:1) c Ξ m,N,j = (cid:16) C mp,N,b , ( π ◦ λ b ) ∗ (cid:16) c g , /mN · c g , “ p − ” /mN (cid:17) (cid:17) . Proof. The first formula is immediate from (5) and the evaluation of the C mp,N,a term above. The secondformula follows by summing the first formula for α = 1 and for α = p − . (cid:3) Evaluation of the norm term. We want to compare the right-hand side of the formula in Corollary3.4.10 with the sum of the c Ξ mp,N,k for all unit liftings k of j . To do this, we shall use the fact that allthe terms c Ξ mp,N,k may be written as the pushforwards of modular units on the same modular curve C mp,N,b . More precisely, if k and ℓ are liftings of j to Z /mp Z which are both units modulo p , we have adiagram H α kℓ ✲ H Y (Γ ( mN ) ∩ Γ ( m ) ∩ U k ) ❄ ∼ = ✲ Y (Γ ( mN ) ∩ Γ ( m ) ∩ U ℓ ) ❄ D m,N,ℓ λ k ❄ ==================== D m,N,k λ ℓ ❄ where α kℓ is any matrix of the form (cid:18) v (cid:19) with v ∈ mN Z congruent to k − ℓ modulo p .Consequently, we can write c Ξ mp,N,k = ( C mp,N,b , ( π ◦ λ b ) ∗ f k ), where f k = α ∗ bk ( ρ k ) ∗ (cid:0) c g , /mpN (cid:1) . We may regard O ( Y (Γ ( mpN ) ∩ Γ ( mp ))) × as a SL ( F p )-module in the obvious way, since Γ ( mpN ) ∩ Γ ( mp ) is the kernel of the surjective reduction map Γ ( mN ) ∩ Γ ( m ) ։ SL ( F p ). With this conventionwe have ( ρ k ) ∗ (cid:0) c g , /mpN (cid:1) = Y u ∈ F × p (cid:18) u − k − ( u − u − ) u (cid:19) ∗ c g , /mpN and thus f k = Y u ∈ F × p (cid:20)(cid:18) u − k − ( u − u − ) u (cid:19) (cid:18) − k − (cid:19)(cid:21) ∗ c g , /mpN = Y u ∈ F × p (cid:18) u − u − k − u − u (cid:19) ∗ c g , /mpN . Let K be the set of possible values of k , i.e. the set of elements of Z /mp Z congruent to j modulo m and not divisible by p . Then as k varies over K , for each fixed u , the expression u − k − u − takes everyvalue in F p exactly once with the exception of u , since k − u − takes every value except 0. So Y k ∈ K f k = Y u,v ∈ F p u =0 v = u c g v /mpN,u /mpN . Here by x and x for x ∈ F p we mean any element of Z /mpN Z congruent to x mod p and to 1 (resp.0) modulo mN . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 21 We find that Y u,v ∈ F p c g v /mpN,u /mpN = c g , /mN (6a) Y v ∈ F p c g v /mpN, /mpN = (cid:18) p (cid:19) ∗ c g ,β/mN (6b) Y u ∈ F p c g u /mpN,u /mpN = (cid:18) (cid:19) ∗ Y u ∈ F p c g ,u /mpN = (cid:18) (cid:19) ∗ (cid:18) p 00 1 (cid:19) ∗ c g , /mN (6c) c g /mpN, /mpN = c g ,β/mN (6d)where β is the inverse of p in Z /mN Z .Combining the above, we have(7) X k ∈ Z /mp Z k = j mod mp ∤ k c Ξ mp,N,k = (cid:16) C mp,N,b , ( π ◦ λ b ) ∗ ( c g , /mN · c g ,β/mN ) (cid:17) − (cid:18) C mp,N,b , ( π ◦ λ b ) ∗ (cid:18) p (cid:19) ∗ c g ,β/mN (cid:19) − (cid:18) C mp,N,b , ( π ◦ λ b ) ∗ (cid:18) (cid:19) ∗ (cid:18) p 00 1 (cid:19) ∗ c g , /mN (cid:19) . Combining the first term on the right-hand side of (7) with Corollary 3.4.8, we see that Theorem 3.4.1is equivalent to Proposition 3.4.11. We have (cid:16) (cid:2) ( h p − i , S ′ p ) + ( S ′ p , h p − i ) (cid:3) σ − p (cid:17) c Ξ m,N,j = (cid:18) C mp,N,b , ( π ◦ λ b ) ∗ (cid:18) p (cid:19) ∗ c g ,β/mN (cid:19) + (cid:18) C mp,N,b , ( π ◦ λ b ) ∗ (cid:18) (cid:19) ∗ (cid:18) p 00 1 (cid:19) ∗ c g , /mN (cid:19) . The first term in Proposition 3.4.11. We now calculate how S ′ p acts on c Ξ m,N,j . We may describethe correspondence S ′ p in terms of the subgroup Γ ( p ) = Γ ( p ) ∩ Γ ( p ); we have S ′ p = ( π ′ ) ∗ ( π ′ ) ∗ , where π ′ and π ′ are the two maps from Y (Γ ( N ) ∩ Γ ( p )) to Y ( N ) given by z pz and z z/p .An application of the strong approximation theorem shows (as usual) that the preimage ( π ′ × − C m,N,j ⊆ Y (Γ ( N ) ∩ Γ ( p )) × Y ( N ) is the single curve F m,N,j given by the set of points of theform (cid:16) zp , z + jm (cid:17) .Applying Lemma 2.4.5 once more, we have a Cartesian square of curves (up to birational equivalence) Y (Γ ( mN ) ∩ Γ ( m ) ∩ Γ ( p )) Y (Γ ( mN ) ∩ Γ ( m )) ✛ p z ← [ z F m,N,j z → (cid:0) z m , p z + j m (cid:1) ✲ C m,N,j ✛ ( p u , v ) ← [ ( u , v ) z → (cid:0) z m , z + j m (cid:1) ✲ The functoriality of pushforward maps gives the following: Proposition 3.4.12. We have ( S ′ p , c Ξ m,N,j,α = (cid:18) ( π ′ × F m,N,j , φ ∗ (cid:18)(cid:18) p 00 1 (cid:19) ∗ c g ,α/mN (cid:19)(cid:19) where φ is the map Y (Γ ( mN ) ∩ Γ ( m ) ∩ Γ ( p )) → ( π ′ × F m,N,j ⊂ Y ( N ) z (cid:16) zmp , pz + jm (cid:17) . We first identify the curve ( π ′ × F m,N,j . Proposition 3.4.13. We have ( π ′ × F m,N,j = (1 × h p i ) − C mp,N,k , for any integer k congruent to p − j modulo m and not divisible by p .More precisely, if k = 1 mod p , and γ ′′ is a suitable element of SL ( Z ) which we shall construct below,then there is a commutative diagram Y (Γ ( mN ) ∩ Γ ( m ) ∩ Γ ( p )) z γ ′′ z ✲ Y (Γ ( mN ) ∩ Γ ( m ) ∩ U k ) Y ( N ) z (cid:16) zmp , pz + jm (cid:17) ❄ (1 × h p i ) ✲ Y ( N ) ,z (cid:16) zmp , z + kmp (cid:17) ❄ where U k is the level p congruence subgroup from the previous section.Proof. We note the following matrix identity, which is easy to verify (although tedious to find): for anyelements p, x, y of a field F , we have (cid:18) yx x − py xy (cid:19) (cid:18) xp (cid:19) p y − px ! = (cid:18) p yp p (cid:19) . In particular, taking F = Q p and x, y ∈ Z × p , we see that the double cosets of SL ( Z p ) in SL ( Q p )generated by (cid:18) x/p (cid:19) and (cid:18) p y/p /p (cid:19) are equal to each other and independent of x and y .Since both (cid:18) p y/p /p (cid:19) and its inverse have entries in p Z p , it follows that (cid:18) x/p (cid:19) γ ∈ SL ( Z p ) (cid:18) p y/p /p (cid:19) for any γ ∈ SL ( Z p ) congruent to (cid:18) − p/x (cid:19) modulo p . (In fact, one can check that it suffices for thematrix to lie in (cid:18) p Z p Z p − p/x + p Z p p Z p (cid:19) .)If x, y are in Z × p ∩ Q , and we choose γ to be in SL ( Z ) and congruent to (cid:18) − p/x (cid:19) modulo p , thenthe matrix γ ′ = (cid:18) x/p (cid:19) γ (cid:18) p y/p /p (cid:19) − will be in SL ( Q ) and will be p -adically integral. If we choose γ to be ℓ -adically close to the identity forsome prime ℓ = p , then γ ′ will be ℓ -adically close to (cid:18) x/p (cid:19) γ (cid:18) p y/p /p (cid:19) − = (cid:18) /p ( px − y ) /p p (cid:19) .So if x, y ∈ Q are units at p and satisfy y = px mod 1, we may choose γ, γ ′ ∈ SL ( Z ) such that: • γ ∈ Γ ( m N ), • γ = (cid:18) − p/x (cid:19) (mod p ), • γ ′ = (cid:18) ∗ ∗ /p (cid:19) (mod N ), • the identity (cid:18) x/p (cid:19) γ = γ ′ (cid:18) p y/p p (cid:19) holds. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 23 We now take y = j/m , and x = k/m for any k congruent to p − j modulo m and invertible modulo p .Then we obtain a commutative diagram H z γz ✲ H Y ( N ) z (cid:0) z, p z + jm (cid:1) ❄ (1 × h p i ) ✲ Y ( N ) z (cid:16) z, z + kmp (cid:17) ❄ or equivalently H z γ ′′ z ✲ H Y ( N ) z (cid:16) zmp , pz + jm (cid:17) ❄ (1 × h p i ) ✲ Y ( N ) z (cid:16) zmp , z + kmp (cid:17) ❄ where γ ′′ = (cid:18) mp 00 1 (cid:19) γ (cid:18) mp 00 1 (cid:19) − . Note that γ ′′ is in Γ ( mN ) ∩ Γ ( mp ), and is congruent modulo p to (cid:18) − /mx (cid:19) = (cid:18) − /k (cid:19) . In the preceding diagram, the left vertical map factors through Y (Γ ( mN ) ∩ Γ ( m ) ∩ Γ ( p )), and γ ′ conjugates this onto Γ ( mN ) ∩ Γ ( m ) ∩ U k ; so we finally obtain the diagram stated in the proposition. (cid:3) Corollary 3.4.14. In the notation of the preceding subsection, we have ( S ′ p , h p − i ) · σ − p · c Ξ m,N,j = (cid:18) C mp,N,b , ( π ◦ λ b ) ∗ (cid:18) (cid:19) ∗ (cid:18) p 00 1 (cid:19) ∗ c g , /mN (cid:19) . Proof. This follows from the previous proposition (and its proof), since the right-hand vertical map inthe diagram of the proposition is the same as λ k above, and ( γ ′′ ) − represents the coset (cid:18) (cid:19) . (cid:3) The second term in Proposition 3.4.11. Now we are left to analyse the operator (1 , S ′ p ). To simplifythe analysis we shall also consider the operator S p given by ( π ′ ) ∗ ( π ′ ) ∗ (rather than S ′ p = ( π ′ ) ∗ ( π ′ ) ∗ );this is the operator associated to the double coset (cid:18) p (cid:19) and is related to S ′ p by the formula S ′ p = h p − i ∗ S p . Again, we find that the preimage ( π ′ ) − C m,N,j in Y ( N ) × Y (Γ ( N ) ∩ Γ ( p )) is a single irreduciblecurve F m,N,j given by points of the form ( z, p ( z + j/m )). Proposition 3.4.15. We have (1 , S p ) ( c Ξ m,N,j,α ) = (cid:18) (1 × π ′ )( F m,N,j ) , φ ∗ (cid:18) p (cid:19) ∗ c g ,α/mN (cid:19) , where the morphism φ is defined by Y (Γ ( mN ) ∩ Γ ( m ) ∩ Γ ( p )) → Y ( N ) z (cid:16) zmp , pz + p jm (cid:17) . Proof. Closely analogous to the previous case. (cid:3) We also have a matrix identity (cid:18) xp (cid:19) (cid:18) − px (cid:19) = (cid:18) x − x p + p yx (cid:19) (cid:18) p py p (cid:19) from which we may deduce that if x, y are rational numbers which are units at p and such that x = py mod 1, there exist γ, γ ′ ∈ SL ( Z ) such that: • γ ∈ Γ ( m N ), • γ = (cid:18) − px (cid:19) (mod p ), • the identity (cid:18) xp (cid:19) γ = γ ′ (cid:18) p py p (cid:19) holds, • γ ′ is congruent to (cid:18) ∗ ∗ p (cid:19) modulo N .Thus the diagram H z γz ✲ H Y ( N ) z ( z, p ( z + y )) ❄ (1 × h p i ) ✲ Y ( N ) z ( z, z + x/p ) ❄ commutes. We take y = j/m , and x = k/m where k is congruent to pj modulo m and not divisible by p . Letting γ ′′ = (cid:18) mp 00 1 (cid:19) γ (cid:18) mp 00 1 (cid:19) − as before, we have the diagram H z γ ′′ z ✲ H Y ( N ) z (cid:16) zmp , pz + p jm (cid:17) ❄ (1 × h p i ) ✲ Y ( N ) z (cid:16) zmp , z + kmp (cid:17) ❄ Again, this shows that (1 × π ′ )( F m,N,j ) = (1 × h p i ) − C mp,N,k . If we choose k to be 1 modulo p , thenthe right vertical map factors through Γ ( mN ) ∩ Γ ( m ) ∩ U k , and the isomorphism between the two isgiven by γ ′′ , which is in Γ ( mN ) ∩ Γ ( p ) and thus acts trivially on (cid:18) p (cid:19) ∗ c g ,α/mN . Thus we have(1 , S p ) σ p · c Ξ m,N,j,α = (1 × h p i ) ∗ (cid:18) C mp,N,b , ( π ◦ λ b ) ∗ (cid:18) p (cid:19) ∗ c g ,α/mN (cid:19) . Taking α = β (the inverse of p modulo mN ), and using the formula c Ξ m,N,j,t = ( h t i , h t i ) σ t c Ξ m,N,j ,we see that (cid:18) C mp,N,b , ( π ◦ λ b ) ∗ (cid:18) p (cid:19) ∗ c g ,β/mN (cid:19) = (1 , h p − i )(1 , S p ) c Ξ m,N,βj,β = (1 , h p − i )(1 , h p i S ′ p ) σ pc Ξ m,N,j,β = (1 , h p − i )(1 , h p i S ′ p ) σ p ( h p − i , h p − i ) σ − p c Ξ m,N,j = ( h p − i , S ′ p ) σ − p c Ξ m,N,j as required, completing the proof of Proposition 3.4.11 and hence of Theorem 3.4.1.3.5. The second norm relation: higher powers of p. We shall also need to know how to calculatenorm p k mm c Ξ p k m,N,j for k = 2 , 3. This is less central to our theory than the k = 1 case, but it willbe needed in order to compare the elements we construct for N coprime to p with their “ p -stabilized”versions. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 25 Theorem 3.5.1. For p ∤ mN we have norm mp mp (cid:0) c Ξ mp ,N,j (cid:1) = ( T ′ p , T ′ p ) c Ξ mp,N,j + (cid:16) p ( h p − i , h p − i ) − ( h p i − , ( T ′ p ) ) − (( T ′ p ) , h p i − )+ 2( h p − i T ′ p , h p − i T ′ p ) σ − p − p ( h p − i , h p − i ) σ − p (cid:17) c Ξ m,N,j and norm mp mp (cid:0) c Ξ mp ,N,j (cid:1) = ( T ′ p , T ′ p ) c Ξ mp ,N,j + (cid:16) p ( h p − i , h p − i ) − ( h p i − , ( T ′ p ) ) − (( T ′ p ) , h p i − ) (cid:17) c Ξ mp,N,j + ( h p − i , h p − i ) (cid:18) T ′ p , T ′ p ) − (( h p i − , ( T ′ p ) ) + (( T ′ p ) , h p i − )) σ − p (cid:19)! c Ξ m,N,j Recall the operator S ′ p which appeared above, satisfying S ′ p = ( T ′ p ) − ( p + 1) h p − i . In terms of theseoperators, the formulae we wish to prove arenorm mp mp (cid:0) c Ξ mp ,N,j (cid:1) = ( T ′ p , T ′ p ) c Ξ mp,N,j + (cid:16) − ( p + 2)( h p − i , h p − i ) − ( h p i − , S ′ p ) − ( S ′ p , h p i − )+ ( h p − i , h p − i ) σ − p (cid:0) T ′ p , T ′ p ) − p ( h p − i , h p − i ) σ − p (cid:1) (cid:17) c Ξ m,N,j andnorm mp mp (cid:0) c Ξ mp ,N,j (cid:1) = ( T ′ p , T ′ p ) c Ξ mp ,N,j − (cid:16) ( p + 2)( h p − i , h p − i ) + ( h p i − , S ′ p ) + ( S ′ p , h p i − ) (cid:17) c Ξ mp,N,j + ( h p − i , h p − i ) (cid:18) T ′ p , T ′ p ) − (cid:2) (2 p + 2)( h p − i , h p − i ) + ( h p i − , S ′ p ) + ( S ′ p , h p i − ) (cid:3) σ − p (cid:19)! c Ξ m,N,j . A routine but unpleasant computation (in which the use of Sage [Sage] was found to be invaluable)shows that Theorem 3.5.1, together with Theorem 3.4.1, implies the following formulae for the norms tolevel prime to p : Theorem 3.5.2. If p ∤ N , we have(a) norm p mm (cid:0) c Ξ p m,N,j (cid:1) = pσ p (cid:16) ( p − − ( h p − i , h p − i ) σ − p ) − (cid:0) ( T ′ p , T ′ p ) σ − p + ( p − (cid:1) P p ( p − σ − p ) (cid:17) (b) norm p mm (cid:0) c Ξ p m,N,j (cid:1) = p σ p (cid:16) ( p − − ( h p − i , h p − i ) σ − p ) − ( p − σ − p ( T ′ p , T ′ p ) + ( p − p − σ − p ( T ′ p , T ′ p ) + ( p − P p ( p − σ − p ) (cid:17) Here P p is the operator-valued Euler factor at p given by P p ( X ) = 1 − ( T ′ p , T ′ p ) X + (cid:16) p (( T ′ p ) , h p − i ) + p ( h p − i , ( T ′ p ) ) − p ( h p − i , h p − i ) (cid:17) X − p ( h p − i T ′ p , h p − i T ′ p ) X + p ( h p − i , h p − i ) X , Evaluation of the ( T ′ p , T ′ p ) term. We begin with a double coset computation in SL ( Q p ). We shallwrite K = SL ( Z p ) and U for the lower-triangular Iwahori subgroup (cid:26)(cid:18) a bc d (cid:19) ∈ K : b ∈ p Z p (cid:27) . Proposition 3.5.3. Let j ≥ . Then the double coset K (cid:18) p − j p j (cid:19) K decomposes as a disjoint union of exactly four double cosets of the Iwahori U , represented by the elements (cid:26)(cid:18) p − j p j (cid:19) , (cid:18) − p − j p j (cid:19) , (cid:18) p j p − j (cid:19) , (cid:18) − p j p − j (cid:19) . (cid:27) Proof. As shown by Iwahori and Matsumoto [IM65, § ( Q p ) = G w ∈ D U wU, where D is the set of matrices of the form (cid:18) p j p j (cid:19) or (cid:18) − p − j p j (cid:19) for some j ∈ Z . Comparingthis with the well-known Cartan decomposition SL ( Q p ) = F j ≥ K (cid:18) p − j p j (cid:19) K gives the statementabove. (cid:3) Proposition 3.5.4. For α ∈ SL ( Q p ) , the index of U ∩ α − U α in U is as follows:(a) p | j | for α ∈ U (cid:18) p j p − j (cid:19) U , j ∈ Z ,(b) p | j +1 | if α ∈ U (cid:18) − p − j p j (cid:19) U , j ∈ Z .Proof. It is clear that the index concerned depends only on the double coset U αU , so we may reduceimmediately to considering the coset representatives in (a) and (b). In each of these cases we find thatthe intersection U ∩ α − U α is a subgroup of the form (cid:26)(cid:18) a bc d (cid:19) ∈ K : p r | b, p s | c (cid:27) for some r ≥ s ≥ 0; this clearly has index p r + s − in U , which gives the above formulae. (cid:3) Corollary 3.5.5. Let j ∈ Z and m ≥ , neither divisible by p , and k ≥ . Then the preimage in Y (Γ ( N ) ∩ Γ ( p )) of the curve C mp k ,N,j ⊆ Y ( N ) is the union of four distinct curves:(1) the curve D given by points of the form (cid:18) z, (cid:18) jmp k (cid:19) z (cid:19) , mapping to C mp k ,N,j with degree p ;(2) the curve D given by points of the form (cid:18) z, γ (cid:18) jmp k (cid:19) z (cid:19) for any γ ∈ Γ ( N ) congruent to (cid:18) ∗∗ ∗ (cid:19) modulo p , again mapping to C mp k ,N,j with degree p ;(3) the curve D given by points of the form (cid:18) z, (cid:18) jmp k (cid:19) γ − z (cid:19) where γ is as before, mapping to C mp k ,N,j with degree p ;(4) the curve D given by points of the form (cid:18) z, γ (cid:18) jmp k (cid:19) γ − z (cid:19) , mapping isomorphically to C mp k ,N,j . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 27 Proof. All of these curves are evidently in the preimage of C mp k ,N,j . One checks that we have (cid:18) jmp k (cid:19) ∈ U (cid:18) − p − k p k (cid:19) Uγ (cid:18) jmp k (cid:19) ∈ U (cid:18) p k p − k (cid:19) U (cid:18) jmp k (cid:19) γ − ∈ U (cid:18) p − k p k (cid:19) Uγ (cid:18) jmp k (cid:19) γ − ∈ U (cid:18) − p k p − k (cid:19) U. Hence the curves D i exhaust the preimage of C mp k ,N,j , by Proposition 3.5.3. The calculation of thedegrees of the maps down follows from Proposition 3.5.4; and since the total degree is ( p + 1) , theymust be distinct. (cid:3) We set α = (cid:18) jmp k (cid:19) , α = γα , α = α γ − , α = γα γ − , so D i is the locus of points of the form ( z, α i z ). Let us define∆ i := (cid:16) π ( D i ) , ( π ) ∗ ( π ) ∗ ( ι ′ mp k ,N,j ) ∗ c g , /mp k N (cid:17) ∈ Z ( Y ( N ) ⊗ Q ( µ mp k , . Then we evidently have ( T ′ p , T ′ p ) c Ξ mp k ,N,j = ∆ + ∆ + ∆ + ∆ . We shall evaluate each of these in turn, showing that D is the norm of c Ξ mp k +1 ,N,j and the remaining∆ i can be calculated in terms of Hecke operators acting on c Ξ mp r ,N,j for r < k .3.5.2. Evaluation of ∆ . Corollary 3.5.6. Pushforward and pullback commute in each of the following four diagrams: (8a) Y (cid:0) Γ ( mp k N ) ∩ Γ ( mp k +1 ) ∩ U (cid:1) z (cid:16) zmp k , z + jmp k (cid:17) ✲ D Y (Γ ( mp k N ) ∩ Γ ( mp k )) ❄ ι ′ mp k ,N,j ✲ C mp k ,N,j π ❄ where U is the subgroup of Γ( p k ) consisting of matrices whose reduction modulo p k +1 lies in the subgroup (cid:18) jm (cid:19) − U (cid:18) jm (cid:19) , and both vertical arrows have degree p ; (8b) Y (cid:0) Γ ( mp k N ) ∩ Γ ( mp k +1 ) (cid:1) z (cid:16) zmp k , γ · z + jmp k (cid:17) ✲ D Y (Γ ( mp k N ) ∩ Γ ( mp k )) ❄ ι ′ mp k ,N,j ✲ C mp k ,N,j ❄ where both vertical arrows have degree p ; (8c) Y (cid:0) Γ ( mp k N ) ∩ Γ ( mp k ) ∩ U (cid:1) z (cid:16) γ · zmp k , z + jmp k (cid:17) ✲ D Y (Γ ( mp k N ) ∩ Γ ( mp k )) ❄ ι ′ mp k ,N,j ✲ C mp k ,N,j ❄ where both vertical arrows again have degree p ; and (8d) Y (cid:0) Γ ( mp k N ) ∩ Γ ( mp k ) (cid:1) z (cid:16) γ · zmp k , γ · z + jmp k (cid:17) ✲ D Y (Γ ( mp k N ) ∩ Γ ( mp k )) ❄ ι ′ mp k ,N,j ✲ C mp k ,N,j ❄ where both vertical arrows are isomorphisms.Proof. Up to conjugation (and identifying C mp k ,N,j and the D i with their normalizations) each diagramtakes the form Y (Γ ∩ Γ ) ✲ Y (Γ ) Y (Γ ) ❄ ✲ Y (Γ) ❄ for subgroups Γ , Γ ⊆ Γ. So it suffices to check in each case that Γ Γ = Γ, or equivalently that[Γ : Γ ] = [Γ : Γ ∩ Γ ]; that is, that the degrees of the two vertical arrows in each diagram are thesame. In each case this reduces to an elementary local computation at p . (cid:3) Proposition 3.5.7 (Evaluation of ∆ ) . We have ∆ = X j ′ ∈ ( Z /mp k +1 Z ) × j ′ = j mod p k c Ξ mp,N,j ′ . Proof. This follows by exactly the same argument as in the case k = 0 considered above. From Corollary3.5.6 we know that the modular unit ( π ) ∗ ( ι ′ mp k ,N,j ) ∗ c g , /mN on D is equal to the pushforward of c g , /mN along the top horizontal arrow in diagram (8a). The subgroups U for all j in a congruenceclass modulo p k are conjugate, and by exactly the same argument as in Proposition 3.4.11 we deducethe result. (cid:3) Evaluation of ∆ and ∆ . We now turn our attention to ∆ . Evidently π ( D ) is the image of Y (Γ ( mp k N ) ∩ Γ ( mp k +1 )) in Y ( N ) under the map z (cid:18)(cid:18) mp k +1 (cid:19) z, (cid:18) p (cid:19) γ (cid:18) j mp k (cid:19) z (cid:19) . Let us write γ = (cid:18) pa bN c d (cid:19) , where d = 1 mod N . Then we find that (cid:18) p (cid:19) γ (cid:18) j mp k (cid:19) = (cid:18) p p (cid:19) (cid:18) a bN c pd (cid:19) (cid:18) j mp k − (cid:19) . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 29 Since scalar matrices act trivially, and (cid:18) a bN c pd (cid:19) acts on Y ( N ) as the diamond operator h p i , we seethat π ( D ) can be written as the image of Y (Γ ( mp k N ) ∩ Γ ( mp k +1 )) ✲ Y ( N ) z ✲ (cid:16) zmp k +1 , h p i · z + jmp k − (cid:17) . This map factors through the natural projection λ : Y (Γ ( mp k N ) ∩ Γ ( mp k +1 )) → Y (Γ ( mp k − N ) ∩ Γ ( mp k +1 ))(indeed, the first component obviously factors through Γ ( N ) ∩ Γ ( mp k +1 ), and the second componentfactors through Y (Γ ( mp k − N ) ∩ Γ ( mp k − )) as the map ι ′ mp k − ,N,j constructed above composed withthe automorphism h p i ). Proposition 3.5.8. We have λ ∗ (cid:0) c g , /mp k N (cid:1) = c g , /mp k − N if k ≥ , c g , /mN · p ! ∗ c g , “ p − ” /mN ! − if k = 1 . (cid:3) We thus have: Proposition 3.5.9. We have ∆ = ( C, φ ) where • C is the image of Y (Γ ( mp k − N ) ∩ Γ ( mp k +1 )) in Y ( N ) under the map β : z (cid:18) zmp k +1 , h p i · z + jmp k − (cid:19) , • φ is the pushforward of c g , /mp k − (resp. of c g , /mN · (cid:18)(cid:18) p (cid:19) ∗ c g , “ p − ” /mN (cid:19) − ) along thismap if k ≥ (resp. if k = 1 ). Evaluation of ∆ . The last, and easiest, term is ∆ . Proposition 3.5.10 (Evaluation of ∆ ) . We have ∆ = p ( h p − i , h p − i ) · ( c Ξ mp k − ,N,j if k ≥ , (cid:0) − ( h p − i , h p − i ) σ − p (cid:1) c Ξ m,N,j if k = 1 .Proof. Contemplating diagram (8d) we know that ∆ is equal to the pushforward of c g , /mp k − N from Y (Γ ( mp k N ) ∩ Γ ( mp k )) to Y ( N ) along the map z (cid:18)(cid:18) p (cid:19) γ (cid:18) mp k (cid:19) z, (cid:18) p (cid:19) γ (cid:18) j mp k (cid:19) z (cid:19) . Since (cid:18) p (cid:19) γ = h p i (cid:18) p 00 1 (cid:19) , this is simply z ( h p i , h p i ) · (cid:18)(cid:18) mp k − (cid:19) z, (cid:18) j mp k − (cid:19) z (cid:19) . This evidently factors as the projection λ : Y (Γ ( mp k N ) ∩ Γ ( mp k )) → Y (Γ ( mp k − N ) ∩ Γ ( mp k − )composed with the map ( h p i , h p i ) ◦ ι ′ mp k − ,N,j .On the other hand, the pushforward of c g , /mp k N from Y (Γ ( mp k N ) ∩ Γ ( mp k )) to Y (Γ ( mp k N ) ∩ Γ ( mp k − )) is clearly (cid:0) c g , /mp k N (cid:1) p , since the degree of the map is p ; and the pushforward from Y (Γ ( mp k N ) ∩ Γ ( mp k − )) to Y (Γ ( mp k − N ) ∩ Γ ( mp k − )) maps c g , /mp k N to c g , /mp k N if k ≥ c g , /mN · (cid:0) c g , “ p − ” /mN (cid:1) − otherwise. (cid:3) Evaluation of ( S ′ p , h p − i ) c Ξ mp k − ,N,j . We now compute the image of c Ξ mp k − ,N,j under the Heckeoperator ( S ′ p , h p i ). Proposition 3.5.11. We have the following coset decompositions in SL ( Q p ) : K (cid:18) j/m (cid:19) K = K = K (cid:18) j/m (cid:19) U ( p ) (where K = SL ( Z p ) ); and K (cid:18) j/mp (cid:19) K = K (cid:18) p − p (cid:19) K = K (cid:18) j/mp (cid:19) U ( p ) ⊔ K (cid:18) p − p (cid:19) U ( p ) ⊔ K (cid:18) p − ξ p (cid:19) U ( p ) ⊔ K (cid:18) p − p (cid:19) U ( p ) . where ξ is any quadratic non-residue in Z × p . Geometrically this is expressed as follows: Proposition 3.5.12. The preimage in Y (Γ ( N ) ∩ Γ ( p )) × Y ( N ) of C mp k − ,N,j is: • if k = 1 , the single curve E consisting of points of the form ( z, z + j/m ) , with degree p ( p + 1) over C m,N,j ; • if k = 2 , the union of four distinct curves E , E , E , E , with degrees ( p, p − , p − , respectivelyover C mp,N,j , where – E is the curve consisting of points of the form ( z, z + j/mp ) , – the curve E is the locus of points of the form ( δ z, z + j/mp ) where δ is any matrix in Γ ( N ) of the form (cid:18) a bc d (cid:19) where p | a and apb − mj is a quadratic residue mod p ; – the curve E is the locus of points of the form ( δ z, z + j/mp ) where δ is any matrix in Γ ( N ) of the form (cid:18) a bc d (cid:19) where p | a and apb − mj is a quadratic nonresidue mod p ; – the curve E is the locus of points of the form ( δ z, z + j/mp ) where δ is any matrix in Γ ( N ) of the form (cid:18) a bc d (cid:19) where p | a and apb − mj = 0 mod p .Proof. This follows from Lemma 2.4.3 and the previous proposition, noting the identity (cid:18) jmp (cid:19) = (cid:18) pmj (cid:19) (cid:18) p ξ p (cid:19) p + pξmj jm − mj ! . (cid:3) Proposition 3.5.13. In the case k = 1 , the image of E in Y ( N ) under ( u, v ) ( u/p , h p i v ) is thecurve C of Proposition 3.5.15 above.In the case k = 2 , the image of E under this map is the curve C ; the images of E and E are both ( h p i , h p i ) C mp,N,j ; and the image of E is ( h p i , h p i ) C m,N, “ p − ” j .Proof. The k = 1 case is clear, as is the assertion for E in case k = 2. The remaining statements are afiddly double coset computation. (cid:3) Proposition 3.5.14. In the case k = 1 , we have ( S ′ p , h p − i ) · c Ξ m,N,j = ( C, β ∗ c g , /mN ) , in the notation of Proposition 3.5.15.In the case k = 2 , we have ( S ′ p , h p − i ) · c Ξ pm,N,j = X i =1 Θ i , where Θ i is the term corresponding to the curve E i of the previous proposition, and Θ = ( C, β ∗ c g , /mpN ) . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 31 Proof. An argument using Lemma 2.4.5 in a familiar manner shows that the pullback of ( ι ′ mp k − ,N,j )( c g , /mp k − )to E coincides with the pushforward of c g , /mp k − along the map Y (Γ ( mp k − N ) ∩ Γ ( mp k +1 )) ✲ Ez ✲ (cid:16) zmp k − , z + jmp k − (cid:17) So the pushforward of this along the map E → Y ( N ) given by ( u, v ) ( u/p , h p i v ) is (cid:0) C, β ∗ c g , /mN (cid:1) ,since the composition of these two maps is β . (cid:3) Combining the preceding proposition with Proposition 3.5.15, we have ∆ = ( S ′ p , h p − i ) · c Ξ m,N,j − ∆ ′ ,where ∆ ′ = C, β ∗ p ! ∗ c g , “ p − ” /mN ! if k = 1,Θ + Θ + Θ if k = 2.We may express the k = 1 case equivalently as∆ ′ = (cid:0) C, β ′∗ c g , “ p − ” /mN (cid:1) where β ′ is the map Y (Γ ( mN ) ∩ Γ ( p ) ∩ Γ ( mp )) → C given by z (cid:16) zmp , h p i · pz + jm (cid:17) .We have seen this map before: we showed above in proposition 3.4.13 that there was a commutativediagram Y (Γ ( mN ) ∩ Γ ( p ) ∩ Γ ( mp )) γ ′′ ✲ Y (Γ ( mN ) ∩ Γ ( m ) ∩ U j ′ ) Y ( N ) z (cid:16) zmp , pz + jm (cid:17) ❄ × h p i ✲ Y ( N ) z (cid:16) zmp , z + j ′ mp (cid:17) ❄ where γ ′′ is a suitable element of Γ ( mN ) (in fact of Γ ( mN ) ∩ Γ ( mp ), although we do not need this), j ′ = p − j mod m is invertible modulo p , and U j ′ is the preimage in a conjugate of the diagonal torus inSL ( F p ). Proposition 3.5.15 (Evaluation of ∆ ′ ) . For k = 1 we have ∆ ′ = ( h p − i , h p − i ) σ − p (cid:0) ( T ′ p , T ′ p ) − σ p − p ( h p − i , h p − i ) σ − p (cid:1) c Ξ m,N,j , and consequently ∆ = (cid:2) ( S ′ p , h p − i ) − ( h p − i , h p − i ) σ − p (cid:0) ( T ′ p , T ′ p ) − σ p − p ( h p − i , h p − i ) σ − p (cid:1)(cid:3) · c Ξ m,N,j . We now consider ∆ . By applying the automorphism of Y ( N ) which switches the two factors, andrunning through essentially the same argument as above, we see that: Proposition 3.5.16 (Evaluation of ∆ ) . For k = 1 we have ∆ = (cid:2) ( h p − i , S ′ p ) − ( h p − i , h p − i ) σ − p (cid:0) ( T ′ p , T ′ p ) − σ p − p ( h p − i , h p − i ) σ − p (cid:1)(cid:3) · c Ξ m,N,j . We now have all the ingredients necessary for the proof in the case k = 1, which will be carried outin § k = 2 there are a few more ingredients we will need.3.5.6. Study of Θ . Let us now consider the term Θ that arises for k = 2. Recall that δ was any elementof Γ ( N ) satisfying a certain congruence modulo p ; we may use strong approximation to make additionalcongruence assumptions modulo primes away from p , so we shall assume that δ = (cid:18) pa bmN c d (cid:19) with ja = mb mod p .For brevity, we shall write Γ( M, N ) for the group (cid:26)(cid:18) a bc d (cid:19) ∈ SL ( Z ) : a = 1 , b = 0 mod Mc = 0 , d = 1 mod N (cid:27) . Proposition 3.5.17. There is a commutative diagram Y (Γ( mp, mpN )) z (cid:18) δ (cid:18) mp (cid:19) z, (cid:18) j mp (cid:19) z (cid:19) ✲ E Y (Γ( mp, mpN )) z εz ❄ ( h p i , h p i ) ι ′ mp,N,j ✲ Y ( N ) ( u, v ) (cid:16) zp , h p i z (cid:17) ❄ where ε is a suitably chosen element of Γ ( N ) and j is the unique integer congruent to j modulo m andto 0 modulo p .Proof. Firstly, we note that Γ ( N ) normalizes Γ( mp, mpN ), so the left-hand vertical arrow is well-defined.More subtly, the well-definedness of the top horizontal arrow follows from the inclusion δ (cid:18) mp (cid:19) Γ( mp, mpN ) (cid:18) mp (cid:19) − δ − ⊆ Γ ( p );indeed δ (cid:18) mp (cid:19) = (cid:18) p m (cid:19) δ ′ where δ ′ = (cid:18) a mbN c d (cid:19) ∈ Γ ( N ) normalizes Γ( mp, mpN ), so δ (cid:18) mp (cid:19) Γ( mp, mpN ) (cid:18) mp (cid:19) − δ − = (cid:18) m p (cid:19) Γ( mp, mpN ) (cid:18) m p (cid:19) − ⊆ Γ ( m N ) ∩ Γ ( p ) . It remains to show that ε may be chosen so that the diagram commutes. We need to choose ε so thatwe have Γ ( N ) h p i (cid:18) mp (cid:19) ε = Γ ( N ) (cid:18) p − p (cid:19) δ (cid:18) mp (cid:19) = Γ ( N ) (cid:18) mp (cid:19) δ ′ , and so that Γ ( N ) (cid:18) j mp (cid:19) ε = Γ ( N ) (cid:18) j mp (cid:19) where j is the unique integer congruent to j mod m and 0 mod p .These conditions are both satisfied if we take ε to be congruent to 1 modulo mN , and to satisfy thesame congruence modulo p as δ ′ , so ε = (cid:18) x jx ∗ ∗ (cid:19) mod p for some x . (cid:3) Corollary 3.5.18. We have Θ = ( h p − i , h p − i ) ∗ (cid:0) C m,N, “ p − ” j , ( ι ′ mp,N,j ) ∗ ( ε − ) ∗ c g , /mpN (cid:1) . Now we shall calculate the pushforward of ( ε − ) ∗ c g , /mpN from Y (Γ( mp, mpN )) to Y (Γ( m, mN )). Proposition 3.5.19. Let α, β ∈ Z be such that α = 0 , β = 1 mod mN and β = 0 mod p . Thenthe pushforward of c g α/mpN,β/mpN from Y (Γ( mp, mpN )) to Y (Γ( m, mN )) along the map z z/p is c g , /mN · (cid:0) c g , “ p − ” /mN (cid:1) − , and hence Θ = ( h p − i , h p − i ) σ p (1 − ( h p − i , h p − i ) σ − p ) c Ξ m,N,j . Proof. A calculation using Theorem 2.2.4 shows that pushing forward to Y (Γ ( mpN ) ∩ Γ ( m )) gives c g α/mN,β/mpN = c g ,β/mpN , and we are now in familiar territory. (cid:3) Study of Θ and Θ . Let δ be any element of Γ ( N ) ∩ Γ ( mN ) whose top left entry is divisibleby p , so δ = (cid:18) pa bmN c d (cid:19) with pa = d = 1 mod N . Let δ ′ = (cid:18) a mbN c pd (cid:19) , so δ ′ ∈ h p i Γ ( N ) and we have δ (cid:18) mp (cid:19) = (cid:18) p m (cid:19) δ ′ .Let E δ be the locus of points in Y (Γ ( N ) ∩ Γ ( p )) × Y ( N ) of the form ( δz, z + j/mp ); this clearlymaps to C mp,N,j under the natural projection to Y ( N ) . We then build the following (rather unwieldy)diagram of modular curves: ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 33 Y (Γ( mp, mpN )) pr ✲ Y ( δ ′ ) − ( m ) − (Γ ( N ) ∩ Γ ( p )) ( m ) δ ′ ∩ (cid:16) j mp (cid:17) − Γ ( N ) (cid:16) j mp (cid:17) (cid:0) p m (cid:1) δ ′ , (cid:16) j mp (cid:17) ∼ = ✲ E δ Y (Γ( mp, mpN ))id ∼ = ❄ pr ✲ Y (cid:0) mp (cid:1) − Γ ( N ) (cid:0) mp (cid:1) ∩ (cid:16) j mp (cid:17) − Γ ( N ) (cid:16) j mp (cid:17) pr ❄ (cid:0) mp (cid:1) , (cid:16) j mp (cid:17) ∼ = ✲ C mp,N,j . pr ❄ Proposition 3.5.20. Suppose that δ ′ = (cid:18) a ′ b ′ c ′ pd (cid:19) with aj − b = 0 mod p . Then the intersection (cid:20) ( δ ′ ) − ( m ) − (Γ ( N ) ∩ Γ ( p )) ( m ) δ ′ ∩ (cid:16) j mp (cid:17) − Γ ( N ) (cid:16) j mp (cid:17)(cid:21) ∩ Γ( m, mN ) consists precisely of those matrices in Γ( m, mN ) which are congruent to ± modulo p .Proof. It suffices to show thatΓ ( N ) ∩ Γ ( p ) ∩ ( m ) δ ′ (cid:16) j mp (cid:17) − Γ ( N ) (cid:16) j mp (cid:17) ( δ ′ ) − ( m ) − consists of matrices that are ± p , since such matrices are clearly preserved under conjugationby ( m ) δ ′ (which is integral at p ).Let γ ∈ Γ ( N ) ∩ Γ ( p ). Then γ is congruent modulo p to (cid:18) x x − (cid:19) for some x ∈ ( Z /p Z ) × . Werequire that (cid:18) j mp (cid:19) ( δ ′ ) − (cid:18) m (cid:19) − γ (cid:18) m (cid:19) δ ′ (cid:18) j mp (cid:19) − ∈ SL ( Z p )or, equivalently, that (cid:18) j (cid:19) ( δ ′ ) − (cid:18) x x − (cid:19) δ ′ (cid:18) j (cid:19) − ∼ = (cid:18) ∗ ∗ ∗ (cid:19) (mod p ) . Substituting the entries of δ ′ , we find that the top right-hand entry of the product on the left is congruentmodulo p to ( aj − b ) cj ( x − x − ). So if aj − b is not divisible by p , then we must have x − x − = 0 mod p ,i.e. x = ± 1, as required. (cid:3) Remark . Conceptually, what is going on here is that we have calculated the intersection of three Borel subgroups of SL ( F p ) in general position relative to each other, which is simply the centre of thegroup. Corollary 3.5.22. The pullback to E δ of ( ι ′ mp,N,j ) ∗ c g , /mpN is equal to the pushforward along the toprow of the above diagram of the modular unit Y γ ∈ U j / {± } γ ∗ c g , /mpN ∈ O ( Y ( mp, mpN ) × ) , where U j is (as above) the torus in SL ( F p ) whose preimage is (cid:18) p (cid:19) − K (cid:18) p (cid:19) ∩ (cid:18) j p (cid:19) − K (cid:18) j p (cid:19) , and we choose a lifting of each element of U/ {± } to an element of Γ( m, mN ) . Note that this depends only rather weakly on δ . We calculated U j explicitly above: it consists of allmatrices of the form (cid:18) u − j − ( u − u − ) u (cid:19) with u ∈ F × p .We now consider the pushforward of this to Y ( N ) along the map ( u, v ) ( h p i − (cid:18) p − p (cid:19) u, v ), sothe image of E δ is one of the components of the image of C mp,N,j under the Hecke operator ( h p i S ′ p , Proposition 3.5.23. The image of E δ under this map is C mp,N,j itself. More specifically, we may find ε ∈ Γ( m, mN ) such that there is a commutative diagram Y (Γ( mp, mpN )) (cid:18) h p i − (cid:18) mp (cid:19) δ ′ , (cid:18) j mp (cid:19)(cid:19) ✲ Y ( N ) Y (Γ( mp, mpN )) ε ∼ = ❄ ι ′ mp,N,j ✲ Y ( N ) id ∼ = ❄ Proof. We must show that ε can be found in such a way that (cid:18) mp (cid:19) δ ′ ∈ Γ ( N ) h p i (cid:18) mp (cid:19) ε and (cid:18) j mp (cid:19) ∈ Γ ( N ) (cid:18) j mp (cid:19) ε. For any ε ∈ Γ( m, mN ), the matrices h p i − (cid:18) mp (cid:19) δ ′ ε − (cid:18) p (cid:19) − and (cid:18) j mp (cid:19) ε − (cid:18) j mp (cid:19) are integral away from p and have bottom right entry congruent to 1 modulo N ; so we need only showthat ε may be chosen such that both are integral at p . So we must show that we can find ε in theintersection ε ∈ (cid:18) j (cid:19) − U ( p ) (cid:18) j (cid:19) ∩ U ( p ) δ ′ . The non-emptiness of this intersection is equivalent to the equality of the double cosets U ( p ) (cid:18) j (cid:19) U ( p ) and U ( p ) (cid:18) j (cid:19) ( δ ′ ) − U ( p ) . However, as we have seen before, there is only one double U ( p ) coset in K other than U ( p ) itself, so thisequality is equivalent to (cid:18) j (cid:19) ( δ ′ ) − / ∈ U ( p ), which is equivalent to our hypothesis b = ja mod p . (cid:3) It remains to be shown that we can choose δ and ε in some reasonable fashion. Let ξ ∈ F × p . Thenwe can take ε = (cid:18) j − (1 − ξ ) − ξ − j − (1 − ξ ) j + ξj − (cid:19) , and δ ′ = (cid:18) j − (1 − ξ ) − ξξ − (cid:19) . A routine verification showsthat (cid:18) j (cid:19) ε (cid:18) − j (cid:19) = (cid:18) j − j − (1 − ξ ) j − (cid:19) is lower-triangular, and that if we take ξ to be aquadratic residue or a nonresidue δ ′ satisfies the congruences stated above, so it suffices to take ξ = 1and one non-square ξ . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 35 Let us write Γ( mp, mpN ) ± for the subgroup of Γ( m, mN ) consisting of matrices that are congruentto ± p . Then we have a diagram Y (Γ( mp, mpN )) σ ✲ Y (Γ( mp, mpN ) ± ) ε ∼ = ✲ Y (Γ( mp, mpN ) ± ) Y (Γ( m, mN )) ✛ (cid:0) p (cid:1) , (cid:16) j p (cid:17) µ Y (Γ( m, mN )) µ (cid:0) p (cid:1) , (cid:16) j p (cid:17) ❄ h p i − (cid:0) p (cid:1) δ ′ , (cid:16) j p (cid:17) µ ✲ Here σ is the natural pushforward map.The images of µ and µ are both given by the curve C of points of the form ( z, z + j/p ) in Y (Γ( m, mN )) , which maps to C mp,N,j under the map ( u, v ) ( u/m, v/m ) to Y ( N ) . We find that µ ∗ ( σ ◦ µ ) ∗ c g , /mpN = Y u ∈ F × p (cid:18) u − j − ( u − u − ) u (cid:19) ∗ c g , /mpN and hence( µ ) ∗ µ ∗ ( σ ◦ µ ) ∗ c g , /mpN = ( µ ) ∗ ( ε − ) ∗ µ ∗ ( σ ◦ µ ) ∗ c g , /mpN = ( µ ) ∗ ( ε − ) ∗ Y u ∈ F × p (cid:18) u − j − ( u − u − ) u (cid:19) ∗ c g , /mpN = Y v ∈ F × p / ± Y u ∈ F × p (cid:20)(cid:18) u − j − ( u − u − ) u (cid:19) ε − (cid:18) v − j − ( v − v − ) v (cid:19)(cid:21) ∗ c g , /mpN Conjugating by (cid:18) j − (cid:19) maps the torus U j onto the diagonal torus and maps ε − onto the matrix (cid:18) j ξ j − (cid:19) , and the above expression becomes Y v ∈ F × p / ± Y u ∈ F × p "(cid:18) − j − (cid:19) (cid:18) u − u (cid:19) (cid:18) j ξ j (cid:19) − (cid:18) v − v (cid:19) (cid:18) j − (cid:19) ∗ c g , /mpN = Y v ∈ F × p / ± Y u ∈ F × p (cid:20)(cid:18) − j − (cid:19) (cid:18) u − v − j u − vξ uvj − (cid:19) (cid:18) j − (cid:19)(cid:21) ∗ c g , /mpN . We may change variables by letting a = uv and b = u − . Then the product becomes Y a ∈ F × p Y b ∈ F × p · ξ (cid:20)(cid:18) − j − (cid:19) (cid:18) j b j − (cid:19) (cid:18) a − a (cid:19) (cid:18) j − (cid:19)(cid:21) ∗ c g , /mpN = ( µ ◦ σ ) ∗ Y b ∈ F × p · ξ (cid:20)(cid:18) − j − (cid:19) (cid:18) j b j − (cid:19) (cid:18) j − (cid:19)(cid:21) ∗ c g , /mpN = ( µ ◦ σ ) ∗ Y b ∈ F × p · ξ (cid:20)(cid:18) j b − j − (1 − b ) (cid:19) (cid:18) j − (cid:19)(cid:21) ∗ c g , /mpN = ( µ ◦ σ ) ∗ (cid:18) j − (cid:19) ∗ Y b ∈ F × p · ξ c g ( − /mpN, ( j − (1 − b )) /mpN Considering Θ and Θ together corresponds to letting b vary over all of F × p . If we were to extend theproduct over all b ∈ F p (residue, nonresidue, or zero), then we would get(9) ( µ ◦ σ ) ∗ (cid:18) j − (cid:19) ∗ (cid:18) p 00 1 (cid:19) ∗ c g α/p, /mN , where α is the image of − /mN in ( Z /p Z ) × (thus α/p = ( − /mN ).The term for b = 0 is just( µ ◦ σ ) ∗ (cid:18) j − j − j − (cid:19) ∗ c g , /mpN = ( µ ◦ σ ) ∗ c g , /mpN , since (cid:18) j − j − j − (cid:19) ∈ U j . This is what we want: it is the definition of c Ξ mp,N,j .What can we say about the expression in (9)? Writing the pushforward in terms of coset representa-tives gives us Y u ∈ F × p "(cid:18) p 00 1 (cid:19)(cid:18) j − (cid:19)(cid:18) j − (cid:19) − (cid:18) u − u (cid:19) (cid:18) j − (cid:19) ∗ c g α/p, /mN = Y u ∈ F × p (cid:20)(cid:18) u − u (cid:19) (cid:18) p 00 1 (cid:19) (cid:18) j − (cid:19)(cid:21) ∗ c g α/p, /mN = (cid:18) j − (cid:19) ∗ (cid:18) p 00 1 (cid:19) ∗ Y u ∈ F × p c g u/p, /mN = (cid:18) j − (cid:19) ∗ c g , /mN · (cid:18)(cid:18) p 00 1 (cid:19) ∗ c g , /mN (cid:19) − ! = c g , /mN · (cid:18)(cid:18) j − (cid:19) ∗ (cid:18) p 00 1 (cid:19) ∗ c g , /mN (cid:19) − The last line is justified by the fact that (cid:18) j − (cid:19) ∗ denotes the action of a matrix congruent to (cid:18) j − (cid:19) modulo p but to the identity modulo mN , and such a matrix will act trivially on c g , /mN .We have seen both of these terms before: the class in CH ( Y ( N ) ⊗ Q ( µ m ) , 1) defined by ( C mp,N,j ,pushforward of c g , /mN ) is (( T ′ p , T ′ p ) − σ p − p h p × p i − σ − p ) c Ξ m,N,j , by Corollary 3.4.10; and the term corre-sponding to ( C mp,N,j , pushforward of (cid:18) j − (cid:19) ∗ (cid:18) p 00 1 (cid:19) ∗ c g , /mN ) is ( h p i − , S ′ p ) c Ξ m,N,j , by Corollary3.4.14.3.5.8. Conclusion of the proof. We can now complete the proof of Theorem 3.5.1 for k = 1.We know that ( T ′ p , T ′ p ) c Ξ mp,N,j = ∆ + ∆ + ∆ + ∆ , and we have shown that: ∆ = norm mp mp (cid:0) c Ξ mp ,N,j (cid:1) (Proposition 3.5.7);∆ = (cid:2) ( S ′ p , h p − i ) − ( h p − i , h p − i ) σ − p (cid:0) ( T ′ p , T ′ p ) − σ p − p ( h p − i , h p − i ) σ − p (cid:1)(cid:3) · c Ξ m,N,j (Proposition 3.5.15);∆ = (cid:2) ( h p − i , S ′ p ) − ( h p − i , h p − i ) σ − p (cid:0) ( T ′ p , T ′ p ) − σ p − p ( h p − i , h p − i ) σ − p (cid:1)(cid:3) · c Ξ m,N,j (Proposition 3.5.16);and ∆ = p ( h p − i , h p − i ) (cid:0) − ( h p − i , h p − i ) σ − p (cid:1) c Ξ m,N,j (Proposition 3.5.10).Combining these statements gives the k = 1 case of Theorem 3.5.1. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 37 In the case k = 2 we have again ( T ′ p , T ′ p ) c Ξ mp ,N,j = ∆ + ∆ + ∆ + ∆ where∆ = norm mp mp (cid:0) c Ξ mp ,N,j (cid:1) (Proposition 3.5.7);∆ = ( S ′ p , h p − i ) · c Ξ mp,N,j − Θ − Θ − Θ ;∆ = ( h p − i , S ′ p ) · c Ξ mp,N,j − Θ ′ − Θ ′ − Θ ′ ;and ∆ = p ( h p − i , h p − i ) · c Ξ mp,N,j (Proposition 3.5.10).Moreover, we haveΘ = ( h p − i , h p − i ) σ p (1 − ( h p − i , h p − i ) σ − p ) c Ξ m,N,j (Proposition 3.5.19),andΘ + Θ = ( h p − i , h p − i ) · c Ξ mp,N,j − (( T ′ p , T ′ p ) − σ p − p h p × p i − σ − p ) c Ξ m,N,j + ( h p i − , S ′ p ) c Ξ m,N,j . The obvious involution of Y ( N ) ⊗ Q ( µ m ) given by swapping the two factors maps c Ξ p k m,N,j to c Ξ p k m,N, − j for each j ; and it interchanges Θ i with Θ ′ i for i = 1 , . . . , 4, so we obtain formulae for theseterms which are identical with the non-primed versions except ( h p − i , S ′ p ) is interchanged with ( S ′ p , h p − i ).Collecting terms gives Theorem 3.5.1 for k = 2.4. Relation to complex L-values Definition of Rankin–Selberg L-functions. We recall the definition of Rankin–Selberg L -functions of pairs of modular forms. Definition 4.1.1. Let f, g be cuspidal new modular eigenforms (of possibly distinct weights k, ℓ andlevels N f , N g ), L a number field containing the coefficients of f and g , and p a prime. We define thelocal Euler factor P p ( f, g, X ) = det (cid:0) − X Frob − p | ( V L λ ( f ) ⊗ V L λ ( g )) I p (cid:1) where λ is an arbitrary place of L of residue characteristic distinct from p , V L λ ( f ) is the L λ -linearrepresentation of G Q attached to f (and similarly for g ) – see § Frob p denotes thearithmetic Frobenius at p . This Euler factor may be defined in purely automorphic terms (cf. [Jac72, Theorem 14.8]), but theabove definition is convenient for our purposes. The following is an elementary calculation: Proposition 4.1.2. If p ∤ N f N g , then P p ( f, g, X ) = (1 − αγX )(1 − αδX )(1 − βγX )(1 − βδX ) where α, β are the roots of X − a p ( f ) X + p k − ε p ( f ) and similarly γ, δ are the roots of X − a p ( g ) X + p ℓ − ε p ( g ) . Completely explicitly, this becomes P p ( f, g, X ) = 1 − a p ( f ) a p ( g ) X + (cid:16) p ℓ − a p ( f ) ε p ( g ) + p k − ε p ( f ) a p ( g ) − p k + ℓ − ε p ( f ) ε p ( g ) (cid:17) X − p k + ℓ − ε p ( f ) a p ( f ) ε p ( g ) a p ( g ) X + p k +2 ℓ − ε p ( f ) ε p ( g ) X . Proposition 4.1.3. We may write P p ( f, g, X ) = Y i =1 (1 − λ i X ) , where each λ i is either 0, or a p -Weil number of weight ≤ ( k + ℓ − . In particular, all poles of themeromorphic function P p ( f, g, p − s ) − have real part at most k + ℓ − .Proof. This is clear from Proposition 4.1.2 if p does not divide the levels of f and g . The remainingcases follow from an explicit computation of the possible local components of f and g , using the Galois-theoretic definition adopted above (since the Weil–Deligne representations attached to f and g must fallinto a finite list of possible types). (cid:3) We now define global Rankin–Selberg L -functions as a product of local terms in the usual way. Definition 4.1.4. We let L ( f, g, s ) = Y p prime P p ( f, g, p − s ) − and for N ≥ we let L ( N ) ( f, g, s ) = Y p prime p ∤ N P p ( f, g, p − s ) − . Proposition 4.1.5. Suppose k ≥ ℓ , and write Γ C ( s ) = (2 π ) − s Γ( s ) . Then the completed L -function Λ( f, g, s ) = Γ C ( s )Γ C ( s − ℓ + 1) L ( f, g, s ) has analytic continuation to all s ∈ C , except for a simple pole at s = k if ℓ = k and f = g ; and itsatisfies a functional equation of the form Λ( f, g, k + ℓ − − s ) = ε ( s ) · Λ( f ⊗ g, s ) where ε is a function of the form Ae Bs for constants A, B .Remark . The function ε ( s ) is, as the notation suggests, a global ε -factor, but we shall not use thisinterpretation here.In particular, if k = ℓ = 2 and s = 1, the value L ( f, g, 1) vanishes (because Γ C ( s − 1) has a simplepole) and we have(10) L ′ ( f, g, 1) = 2 π Λ( f, g, . Real-analytic Eisenstein series. We now express the Rankin–Selberg L -function in terms of thePetersson product with a non-holomorphic Eisenstein series, the original example of the Rankin–Selbergmethod. Definition 4.2.1. Let k ≥ ∈ Z , and α ∈ Q / Z .(1) For τ ∈ H , s ∈ C with k + 2 ℜ ( s ) > , we define E ( k ) α ( τ, s ) = ( − πi ) − k π − s Γ( s + k ) X ′ ( m,n ) ∈ Z ℑ ( τ ) s ( mτ + n + α ) k | mτ + n + α | s , where the prime denotes that the term ( m, n ) = (0 , is omitted if α = 0 (but not otherwise).(2) For τ, s as above, define F ( k ) α ( τ, s ) = ( − πi ) − k Γ( s + k ) π − s X ′′ ( m,n ) ∈ Z e πiαm ℑ ( τ ) s ( mτ + n ) k | mτ + n | s where the double prime denotes that the term ( m, n ) = (0 , is omitted (always). Proposition 4.2.2. The above series have the following properties:(i) (Automorpy) If N α = 0 , then for fixed τ both E ( k ) α and F ( k ) α are preserved by the weight k actionof Γ ( N ) , and moreover the diamond operators act on α by multiplication in the obvious way.(ii) (Action of Atkin–Lehner involutions) If N α = 0 , then we have F ( k ) α ( τ, s ) = N − k − s X x ∈ Z /N Z e πiαx τ − k E ( k ) x/N (cid:0) − Nτ , s (cid:1) . (iii) (Differential operators) The Maass–Shimura weight-raising differential operator δ k := 12 πi (cid:18) dd τ + kτ − τ (cid:19) (cf. [Shi76, Equation (2.8)] ) acts on the Eisenstein series via δ k E ( k ) α ( τ, s ) = E ( k +2) α ( τ, s − and similarly for F ( k ) α . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 39 (iv) (Analytic continuation and functional equation) For fixed k, τ, α , both functions E ( k ) α ( τ, s ) and F ( k ) α ( τ, s ) have meromorphic continuations to the whole s -plane, which are holomorphic everywhereif k = 0 ; and we have E ( k ) α ( τ, s ) = F ( k ) α ( τ, − k − s ) . (v) (Relation to Siegel units) We have E (0) α ( τ, 0) = 2 log | g ,α ( τ ) | where g ,α is the Siegel unit of § Parts (i)–(iii) are easy explicit computations. Part (iv) is a standard application of the Poissonsummation formula, and (v) is formula (3.8.4)(iii) of [Kat04]. (cid:3) Now let f, g be any two newforms of levels N f , N g dividing N , and weights k, ℓ , with k > ℓ . Let˘ f ∈ S k (Γ ( N ))[ π f ] and ˘ g ∈ S ℓ (Γ ( N ))[ π g ] be forms in the oldspaces at level N attached to f and g (which we shall think of as “test vectors”). For any α ∈ N Z / Z , set D ( ˘ f , ˘ g, x, s ) = Z Γ ( N ) \H ˘ f ( − τ ) ˘ g ( τ ) E ( k − ℓ ) α ( τ, s − k + 1) ℑ ( τ ) k − d x d y = D ˘ f ∗ ( τ ) , ˘ g ( τ ) · E ( k − ℓ ) α ( τ, s − k + 1) E Γ ( N ) . The next theorem shows that the function D ( ˘ f , ˘ g, /N, s ) is an “approximation” to the completed L -function Λ( f, g, s ) of the previous section, differing from it only by possible bad Euler factors at primes ℓ | N . Theorem 4.2.3 (Rankin–Selberg, Shimura) . We have D ( ˘ f , ˘ g, /N, s ) = 2 − k i k − ℓ N s +2 − k − ℓ Λ( f, g, s ) C ( ˘ f , ˘ g, s ) , where C ( ˘ f , ˘ g, s ) := Y p | N P p ( f, g, p − s ) X n ∈ S ( N ) a n ( ˘ f ) a n (˘ g ) n − s is a polynomial in the variables p − s for p | N ; in particular, it is holomorphic for all s ∈ C . Here S ( N ) is the set of integers all of whose prime factors divide N .Proof. See [Kat04, Proposition 7.1]; our E ( j )1 /N ( τ, s ) corresponds to( − πi ) − j Γ( s + j ) π − s ℑ ( τ ) s E ( j, τ, /N, s )in Kato’s notation, where j = k − ℓ . To see that C ( ˘ f , ˘ g, s ) is a polynomial, it suffices to consider thecase when ˘ f = f ( az ) and ˘ g = g ( bz ) for integers a | N/N f , b | N/N g , in which case the result is clear. (cid:3) In particular, for s = 1 and k = ℓ = 2, using Equation (10) and the above proposition gives(11) D ( ˘ f , ˘ g, /N, 1) = (4 π ) − L ′ ( f, g, C ( ˘ f , ˘ g, . Remark . If f , g have coprime levels N f , N g with N f N g = N , and we take ˘ f = f and ˘ g = g to bethe normalized newforms, then C ( ˘ f , ˘ g, s ) is identically 1, so in this case D ( ˘ f , ˘ g, /N, s ) is N s Λ( f, g, s )up to constants.From the functional equation for the real-analytic Eisenstein series, and the action of Atkin–Lehnerinvolutions, we have(12) D ( ˘ f , ˘ g, x, k + ℓ − − s ) = D ˘ f ∗ ( τ ) , ˘ g ( τ ) · F ( k − ℓ ) x/N ( τ, s − k + 1) E Γ ( N ) = N − s X y ∈ Z /N Z e πixy D ( w N ˘ f , w N ˘ g, y, s ) . Here w N ˘ f is the function τ N − τ − k ˘ f ( − / ( N τ )); that is, we have chosen our normalizations so that w N is an involution in weight 2 (but not in more general weights). The Beilinson regulator. For any smooth variety X over a subfield of C there is a canonicalmap, the Beilinson regulator , from H M ( X, Z (2)) into complex-analytic Deligne–Beilinson cohomology.These maps were introduced in [Be˘ı84]. We shall only need these maps for H M ( X, Z (2)) where X is aprojective surface, in which case the target group can be identified with de Rham cohomology: Theorem 4.3.1 (Beilinson, cf. [Jan88b, p. 45]) . Let X be a smooth projective surface over C (or asubfield of C ). There is a homomorphism reg C : CH ( X, → H ( X/ C ) / Fil = (cid:0) Fil H ( X/ C ) (cid:1) ∨ which sends the class of P j ( Z j , g j ) ∈ Z ( X, to the linear functional (13) ω πi X j Z Z j − Z sing j ω log | g j | . We now show that the images of the generalized Beilinson–Flach elements Ξ m,N,j under reg C , pairedwith differentials corresponding to weight 2 modular forms f, g , are related to the derivatives of Rankin–Selberg L -functions at the point s = 1. More precisely, we shall apply reg C to a lifting of Ξ m,N,j toCH ( X ( N ) ⊗ Q ( µ m ) , ⊗ Q ; the result will turn out to be independent of the choice of lifting. Definition 4.3.2. If f ∈ S (Γ ( N )) , we let f ∗ ∈ S (Γ ( N )) be the form obtained by applying complexconjugation to the Fourier coefficients of f .We let ω f denote the holomorphic differential on X ( N ) whose pullback to H is πif ( z ) d z , and η ah f the anti-holomorphic differential ω f ∗ , whose pullback is − πif ( − z ) d z .Remark . (1) The factor 2 πi is convenient since d qq = 2 πi d z .(2) The map f → η ah f is C -linear and Hecke-equivariant (whereas the more obvious map f ω f hasneither of these desirable properties). Theorem 4.3.4 (Beilinson, cf. [BDR12, Proposition 4.1]) . Let e Ξ N be any element of CH ( X ( N ) , lifting Ξ N := Ξ ,N, ∈ CH ( X ( N ) , , and let p , p be the projections of X ( N ) onto its two factors.Then for ˘ f , ˘ g as above we have D reg C (cid:16)e Ξ N (cid:17) , p ∗ ( η ah˘ f ) ∧ p ∗ ( ω ˘ g ) E = 2 π D ( ˘ f , ˘ g, /N, 1) = L ′ ( f, g, C ( ˘ f , ˘ g, . Proof. We have D ′ ( f, g, /N, 1) = Z Γ ( N ) \H f ( − ¯ τ ) g ( τ ) E (0)1 /N ( τ, 0) d x ∧ d y ! = 2 Z Γ ( N ) \H f ( − τ ) g ( τ ) log (cid:12)(cid:12) g , /N ( τ ) (cid:12)(cid:12) d x ∧ d y = 2 Z Γ ( N ) \H f ( − τ ) g ( τ ) log (cid:12)(cid:12) g , /N ( τ ) (cid:12)(cid:12) ( − πi d¯ z ) ∧ (2 πi d z )8 π i = 12 π πi Z Y ( N )( C ) log (cid:12)(cid:12) g , /N (cid:12)(cid:12) η ah f ∧ η g ! . We compare this with Beilinson’s formula for the regulator on CH ( X ( N ) , 1) (Theorem 4.3.1). Weknow that e Ξ N can be written as the class of (∆ , g , /N ) (where ∆ = C ,N, is the diagonal in X ( N ) )plus a linear combination of elements supported on cuspidal components. It is clear that p ∗ ( η ah f ) ∧ p ∗ ( ω g )restricts to 0 on any horizontal or vertical component, and to η ah f ∧ η g on ∆; so we obtain D reg C (cid:16)e Ξ N (cid:17) , η ah f ∧ η g E = 12 πi Z Y ( N ) log (cid:12)(cid:12) g , /N (cid:12)(cid:12) η ah f ∧ η g = 2 πD ′ ( f, g, (cid:3) We are interested in a version of Theorem 4.3.4 for m ≥ 1, incorporating twists by Dirichlet characters.This relation becomes easier to state if we introduce “equivariant” versions of some of our objects, asfollows: ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 41 Definition 4.3.5. For N ≥ , m ≥ as above, and cusp forms f, g of level N which are eigenforms forthe Hecke operators away from N , we define the following elements: g m = X a ∈ ( Z /m Z ) × [ a ] − ⊗ g (cid:0) z + am (cid:1) ∈ C [( Z /m Z ) × ] ⊗ C S (Γ ( m N ) , C ) and the C [( Z /m Z ) × ] -valued Dirichlet series L ( mN ) ( f, g, ( Z /m Z ) × , s ) = Y ℓ ∤ mN P ℓ ( f, g, [ ℓ ] ℓ − s ) − . (There is no obvious way to define an equivariant Euler factor at the primes dividing m .) Proposition 4.3.6. (a) We have a n ( g m ) = a n ( g ) τ ( n, m ) , where τ ( n, m ) is the “universal Gauss sum” P a ∈ ( Z /m Z ) × [ a ] − e πina/m ∈ C [( Z /m Z ) × ] .(b) If we extend the Hecke operators on S (Γ ( N )) linearly to C [( Z /m Z ) × ] ⊗ C S (Γ ( m N )) , then wehave T n ( g m ) = [ n ] t g ( n ) g m h n i ( g m ) = [ n ] ε g ( m ) g m for all n such that ( n, mN ) = 1 , where t g ( n ) and ε g ( n ) are the eigenvalues of g for the T n and h n i operators respectively. (We can interpret (b) above as stating that g m transforms under the Hecke operators away from mN as “ g twisted by the universal character of level m ”.) Proof. Part (a) is immediate by a q -expansion computation. For part (b), we note that the statementregarding the diamond operators can be verified directly – by essentially the same computation asProposition 2.7.5(4) – and the statement for the T n ’s now follows immediately from the standard formulaefor the action of T n on q -expansions, together with the easily verified fact that τ ( nn ′ , m ) = [ n ] τ ( n ′ , m )if ( n, m ) = 1. (cid:3) We extend the Beilinson regulator to a homomorphismreg C [( Z /m Z ) × ] : CH ( X ( N ) ⊗ Q ( µ m ) , → C [( Z /m Z ) × ] ⊗ C (cid:0) Fil H ( X/ C ) (cid:1) ∨ by mapping δ to P a ∈ ( Z /N Z ) × [ a ] ⊗ reg C ( σ a · δ ). (Note that this is not a homomorphism of modules overthe group ring C [( Z /m Z ) × ]; the Poincar´e duality pairing interchanges the natural action of C [( Z /m Z ) × ]with its inverse.) Theorem 4.3.7. Let ˘ f , ˘ g be as above, and let e Ξ m,N, be any lifting of Ξ m,N, to X ( N ) . Then aselements of C [( Z /m Z ) × ] we have D reg C [( Z /m Z ) × ] ( e Ξ m,N, ) , p ∗ ( η ah˘ f ) ∧ p ∗ ( ω ˘ g ) E = L ′ ( mN ) ( f, g, ( Z /m Z ) × , A ( ˘ f , ˘ g, m, , where we define A ( ˘ f , ˘ g, m, s ) = X a ∈ ( Z /m Z ) × [ a ] − X n ∈ S ( mN ) a n ( f ) a n ( g ) e πian/m n − s . Proof. As we showed in the previous section, e Ξ m,N,j may be represented as the class of an element in Z ( X ( N ) ⊗ Q ( µ m ) , ⊗ Q [( Z /m Z ) × ] which differs by negiligible elements from (cid:0) C m,N,j , ( ι m,N,j ) ∗ ( g , /m N ) (cid:1) . As in the case m = 1 considered above, these negligible elements pair to 0 with the differential p ∗ ( η ah f ) ∧ p ∗ ( ω g ). Hence we have h reg C ( ^ Ξ m,N ) , p ∗ ( η ah f ) ∧ p ∗ ( ω g ) i = X j ∈ ( Z /m Z ) × [ j ] − Z C m,N,j log (cid:12)(cid:12) ( ι m,N,j ) ∗ ( g , /m N ) (cid:12)(cid:12) · p ∗ ( η ah f ) ∧ p ∗ ( ω g )= X j ∈ ( Z /m Z ) × [ j ] − Z X ( m N ) log (cid:12)(cid:12) g , /m N (cid:12)(cid:12) · ( p ◦ ι m,N,j ) ∗ ( η ah f ) ∧ ( p ◦ ι m,N,j ) ∗ ( ω g ) . By construction p ◦ ι m,N,j is just the natural projection map X ( m N ) → X ( N ), so the pullback of η ah f along this map is just η ah f again (where now we consider f as a modular form of level m N ). On theother hand, p ◦ ι m,N,j corresponds to the map z z + jm on the upper half-plane, so ( p ◦ ι m,N,j ) ∗ ( ω g )is the differential whose pullback to H is 2 πig (cid:0) z + jm (cid:1) d z , and hence we have X j [ j ] − ( p ◦ ι m,N,j ) ∗ ( ω g ) = 2 πi g m ( z ) d z as elements of C [( Z /m Z ) × ] ⊗ Ω ( X ( m N )). Hence, by exactly the same computation as above, h reg C ( ^ Ξ m,N ) , p ∗ ( η ah f ) ∧ p ∗ ( ω g ) i = 4 π Z X ( m N ) f ( − ¯ τ ) g m ( τ ) log (cid:12)(cid:12) g , /m N (cid:12)(cid:12) d x ∧ d y. As remarked above, g m is an eigenform for the Hecke operators away from mN ; so we may now applyexactly the same formal manipulations as in the proof of [Kat04, Proposition 7.1], but with group ringcoefficients rather than C coefficients, and the result follows in this case also. (cid:3) A non-vanishing result. In this section, we shall use the results of the previous section, togetherwith a deep theorem of Shahidi on the non-vanishing of Rankin–Selberg L -values, to show that theelements Ξ m,N,j are not all zero (which is in no way obvious from their construction). Theorem 4.4.1 (Shahidi, [Sha81, Theorem 5.2]) . Let f, g be any two newforms of weight 2. Then thecompleted L -function Λ( f, g, s ) is holomorphic and nonvanishing on the line ℜ ( s ) = 2 , unless f = g ∗ , inwhich case it has a simple pole at s = 2 .Remark . We have stated only a special case of Shahidi’s very general theorem, which applies toautomorphic forms on GL n × GL m over an arbitrary number field. Note also that Shahidi’s normaliza-tions are slightly different from ours (he normalizes the L -function so that the abcissa of symmetry is s = , independently of the weights of f and g , while we normalize it to be at s = k + ℓ − = ). Corollary 4.4.3. If Σ is a finite set of primes, then the function L Σ ( f, g, s ) has a zero at s = 1 of order r + r , where r = ( if f ∗ = g if f ∗ = g and r is the sum of the orders of the poles at s = 1 of the Euler factors L p ( f, g, s ) for primes p ∈ Σ .Proof. Applying the functional equation for the completed L -function, which switches s with 3 − s , wededuce from Shahidi’s result that that Λ( f, g, s ) is holomorphic and nonvanishing (resp. has a simplepole) at s = 1 if f = g ∗ (resp. if f = g ∗ ).However, the L -factor at ∞ , L ∞ ( f, g, s ) = Γ C ( s )Γ C ( s − s = 1, so the order ofvanishing of L ( f, g, s ) is r as defined above. Since L Σ ( f, g, s ) is L ( f, g, s ) divided by the product of the L -factors at primes in Σ, the result clearly follows. (cid:3) Remark . Note that if f ∗ = g , then the local L -factor vanishes at s = 1 for every prime, so r willtend to be rather large in this case. Corollary 4.4.5. Let f, g be any two newforms, Σ any set of primes, and p any prime in Σ . Then forall but finitely many Dirichlet characters χ of p -power conductor, Q ℓ ∈ Σ L ℓ ( f, g ⊗ χ, s ) is holomorphicand nonzero at s = 1 .Proof. If χ has sufficiently large p -power conductor, then the local L -factor of f ⊗ g ⊗ χ at p is identically1; so it suffices to consider the L -factors at primes ℓ = p . However, since χ has conductor prime to ℓ , L ℓ ( f ⊗ g ⊗ χ, s ) = P ℓ ( χ ( ℓ ) ℓ − s ) − , so it suffices to arrange that χ ( ℓ ) ℓ − does not lie in the finite set ofzeroes of the polynomial L ℓ ( f ⊗ g, X ). It is clear that this may also be achieved by ensuring that theconductor of χ is sufficiently big. (cid:3) Corollary 4.4.6. Given any two forms f, g of level N that are eigenvectors for all Hecke operators,and p any prime, there is k ≥ such that the projection of Ξ mp k ,N, to the ( f, g ) -isotypical quotient of CH ( Y ( N ) ⊗ Q ( µ mp k ) , is nonzero.Proof. Immediate from the previous corollary and Theorem 4.3.7. (cid:3) ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 43 Relation to p -adic L -values In this section we develop an analogue of the m = 1 case of Theorem 4.3.7 in the p -adic setting. Thisis essentially a variant of the main theorem of [BDR12].5.1. Holomorphic Eisenstein series. We begin by constructing some holomorphic Eisenstein serieswhich may be defined over a number field. We follow chapter 3 of [Kat04] closely, but we work on Y ( N )rather than Y ( N ). Our purpose is to define, for α ∈ Q / Z , the following modular forms: • E ( k ) α ∈ M k (Γ ( N )), where k ≥ k = 2; • e E (2) α ∈ M (Γ ( N )); • F ( k ) α ∈ M k (Γ ( N )), for k ≥ 1, with α = 0 if k = 2.We set E ( k ) α ( τ ) = E ( k ) α ( τ, , and similarly for F ( k ) . Proposition 5.1.1. If k ≥ , k = 2 , then E ( k ) α , F ( k ) α ∈ M k (Γ ( N )) for any α ∈ N Z / Z .For k = 2 , we have F (2) α ∈ M (Γ ( N )) for any α = 0 , and e E (2) α := E (2) α − E (2)0 ∈ M (Γ ( N )) (for any α ). The function F (2)0 = E (2)0 is a C ∞ function on H invariant under the weight 2 action of Γ ( N ) , withslow growth at the cusps, but is not holomorphic.Proof. See [Kat04, § E ( k ) α is Kato’s E ( k )0 ,α . (cid:3) We have q -expansion formulae for both families. Let α ∈ Q / Z . For ℜ ( s ) > 1, define ζ ( α, s ) = X n ∈ Q ,n> n = α mod Z n − s and ζ ∗ ( α, s ) = ∞ X n =1 e πiαn n − s as in [Kat04, § ζ ( α, s ) and ζ ∗ ( α, s ) have meromorphic continuation to all s ∈ C , andsatisfy ζ ∗ ( α, − s ) = Γ( s )(2 π ) s (cid:16) e − iπs/ ζ ( − α, s ) + e iπs/ ζ ( α, s ) (cid:17) , a version of the standard functional equation for the Hurwitz zeta function. Proposition 5.1.2. Let k ≥ , α ∈ Q / Z .(1) Assume k = 2 . Then we have E ( k ) α = a + X n ≥ X d | n d k − ( e πiαd + ( − k e − πiαd ) q n , where a = ( ζ ∗ ( α, − k ) if k ≥ ( ζ ∗ ( α, − ζ ∗ ( − α, if k = 1 . . (2) We have e E (2) α = a + X n ≥ X d | n d ( e πiαd + e − πiαd − q n , where a = ζ ∗ ( α, − 1) + .(3) Assume α = 0 in the case k = 2 . Then F ( k ) α = a + X n ≥ X d | n (cid:0) nd (cid:1) k − ( e πiαd + ( − k e − πiαd ) q n , where a = ( ζ (1 − k ) if k ≥ ( ζ ∗ ( α, − ζ ∗ ( − α, if k = 1 . . Proof. This is [Kat04, Proposition 3.10]. Note that there is a typographical error in the statement ofthe proposition loc.cit. ; there is an extra star in the formula for P a n n − s in case (1), and the formulashould read X n ∈ Q ,n> a n n − s = ζ ( α, s ) ζ ∗ ( β, s − k + 1) + ( − k ζ ( − α, s ) ζ ∗ ( − β, s − k + 1) . (cid:3) Nearly holomorphic modular forms. For k ≥ 0, we define (following e.g. [Shi86, Shi00]) thespace of nearly holomorphic modular forms M nh k (Γ ( N ) , C ). This is the space of C ∞ slowly-increasingfunctions on H which are invariant under the weight k action of Γ ( N ) and are annihilated by somepower of the Maass–Shimura weight-lowering differential operator ε k = − πi ℑ ( τ ) dd τ . Any such function is in fact annhilated by ε [ k/ , and can be expanded as a finite sum(14) f ( τ ) = [ k/ X j =0 f j ( τ ) ( π ℑ ( τ )) − j , where the f j are holomorphic functions. (In particular, any nearly holomorphic form of weight 0 or 1 isin fact a holomorphic form.) For K a number field containing the N -th roots of unity we shall say that f ∈ M nh k (Γ ( N ) , K ) is defined over K if the Fourier coefficients of the holomorphic functions f j are in K , and write M nh k (Γ ( N ) , K ) for the space of such functions.We let S nh k (Γ ( N ) , K ) be the subspace of rapidly decreasing functions in M nh k (Γ ( N ) , K ). If k ≥ S nh k (Γ ( N ) , K ) coincides with the space defined algebraically in [DR12, § Corollary 5.2.1. Let k, j be integers with k ≥ and j ∈ [0 , k − . Then for any α ∈ N Z / Z , thefunction τ E ( k ) α ( τ, − j ) lies in M nh k (Γ ( N ) , Q ( µ N )) .Proof. We first note that E ( k ) α , F ( k ) α ∈ M nh k (Γ ( N ) , K ) for all k ≥ 1. This is clear for k = 1 or k ≥ k = 2 it suffices to check that E (2)0 = F (2)0 is nearly holomorphic, which is clear from the formula E (2)0 ( τ ) = 14 π ℑ ( τ ) − + 2 X n ≥ X d | n d q n . With this in hand, we obtain the near-holomorphy of E ( k ) α ( τ, − j ) for 0 ≤ j ≤ k − by applying δ j to E ( k − j ) α = E ( k − j ) α ( τ, F ( k − j ) α shows that F ( k ) α ( τ, − j ) is nearlyholomorphic for j in the same range; but F ( k ) α ( τ, − j ) = E ( k ) α ( τ, − k + j ) by the functional equation, asrequired, so we obtain the result for all j ∈ [0 , k − (cid:3) We define the q -expansion of a nearly-holomorphic modular form f to be the Fourier expansion of theholomorphic Z -periodic function f , when we write f in the form (14). Then δ corresponds to q dd q on q -expansions, and the following is clear from Proposition 5.1.2 and the proof of the previous proposition: Proposition 5.2.2. For any j ∈ [0 , k − , the q -expansion of the nearly-holomorphic form E ( k ) α ( τ, − j ) is a + X n ≥ X d | n d k − − j (cid:0) nd (cid:1) j ( e πiαd + ( − k e − πiαd ) q n , where a = 0 unless j ∈ { , k − } . This is for compatibility with our notation for classical modular forms, since in our model of Y ( N ), the cusp ∞ is notrational. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 45 P-adic families of Eisenstein series. We now use the q -expansion formulae above as motivationfor defining a two-parameter family of p -adic Eisenstein series, which can be regarded as an analogue ofthe E ( k ) ( − , s ), with two continuous p -adic parameters φ , φ replacing the discrete parameter k and thecontinuous real-analytic parameter s . Definition 5.3.1. Choose some (sufficiently large) finite extension L/ Q p and let Λ = O L [[ Z × p ]] , theIwasawa algebra of Z × p . Let Ω = Spf Λ , the weight space classifying continuous characters φ : Z × p → C p ;and consider the formal power series with coefficients in Λ b ⊗ Λ given by E α ( φ , φ ) = X n ≥ p ∤ n X d | n φ ( d ) φ (cid:0) nd (cid:1) (cid:2) e πiαd + εe − πiαd (cid:3) q n ∈ O L [[ q ]] , where ε = − φ ( − φ ( − .Note . We consider Z × Z as a subset of Ω × Ω in the natural way. Then for k ≥ , k = 2, we have E α ( k − , 0) = ( E ( k ) α ) [ p ] and E α (0 , k − 1) = ( F ( k ) α ) [ p ] , where ( − ) [ p ] denotes the “ p -depletion” operator.We now fix a newform g ∈ S ℓ (Γ ( N g )), for some ℓ and some N g | N . (Although the weight ℓ will befixed in our discussion, it is convenient to keep it in the notation as a parameter, since this will makeour notation more consistent with [BDR12].) We will write ˘ g for any element of S ℓ ( N )[ π g ]. Definition 5.3.3. For integers k, j , we define Ξ( k, ℓ, α, j ) ord ,p = e ord [ E α ( j − ℓ, k − − j ) · ˘ g ] . This is an ordinary Λ-adic family of modular forms, parametrized by k and j (we are taking ℓ tobe fixed here, in order to avoid the need to make any ordinarity hypotheses on g ). For any ( k, j ),Ξ( k, ℓ, α, j ) ord ,p is a p -adic modular form of weight k .We now compare this with the complex-analytic theory. It is clear that E α ( j − ℓ, k − − j ) is the p -depletion of the nearly-overconvergent form τ E ( k − ℓ ) ( τ, − k + j + 1). Definition 5.3.4. For ℓ ≤ j ≤ k − , let Ξ( k, ℓ, α, j ) denote the nearly-holomorphic modular form ofweight k − ℓ given by Ξ( k, ℓ, α, j )( τ ) = E ( k − ℓ ) α ( τ, − k + j + 1) · ˘ g, and Ξ( k, ℓ, α, j ) hol its image under the holomorphic projector.Notation. Let H ( Y ( N ) , L k − , ∇ ) denote the de Rham cohomology of Y ( N ) with coefficients in the( k − Y ( N ), endowed with its Gauss–Manin connection. Proposition 5.3.5. Let k, ℓ be fixed, with k > ℓ . Let f be a newform in S k (Γ ( N f )) , for some N f | N ,and let e f ∗ be the projection to the f ∗ -isotypic component in the Hecke algebra acting on S k (Γ ( N )) . As-sume f is ordinary at p , and let j ∈ [ ℓ, k − . Then we have the following relation in H ( Y ( N ) , L k − , ∇ ) : e f ∗ Ξ( k, ℓ, α, j ) ord ,p = E ( f, g, j ) E ( f ) e f ∗ e ord Ξ( k, ℓ, α, j ) hol , where E ( f ) = 1 − p − β p ( f ) α p ( f ) − and E ( f, g, j ) = (1 − p − j β p ( f ) α p ( g ))(1 − p − j β p ( f ) β p ( g )) × (1 − p j − α p ( f ) − α p ( g ) − )(1 − p j − α p ( f ) − β p ( g ) − ) . Here α f , β f are the roots of the Hecke polynomial of f at p , and similarly for g .Proof. This follows from Proposition 4.15 of [DR12] with the f, g, h of the theorem taken to be f , E ( k − ℓ ) α ( − , − k + j + 1) and g . (Note that the special case j ≥ k + ℓ − is [BDR12, Proposition 2.7].) (cid:3) Interpolation in Hida families. We now interpolate the left-hand side of Proposition 5.3.5 inHida families. Notation. Let f be a newform (of some level N f | N ) and let f be the Hida family through f (withcoefficients in some finite flat Λ-algebra Λ f ). Then we define the space S ord ( N ; Λ f )[ π f ] for the Λ f -moduleof families of oldforms at level N corresponding to f , which is simply the space of formal q -expansionsspanned over Λ f by f ( q d ) for d | N/N f . We write ˘ f for a generic element of S ord ( N ; Λ f )[ π f ], which weshall think of as a “test vector” associated to f .We shall continue to write g for a newform in S ℓ ( N g ) for some N g | N , and ˘ g for a generic element of S ℓ ( N ; K )[ π g ]. Proposition 5.4.1. For any ˘ f ∈ S ord ( N ; Λ f )[ π f ] and ˘ g ∈ S ℓ ( N ; K )[ π g ] as above, and any α ∈ N Z / Z ,there exists an element D p (˘ f , ˘ g, α ) ∈ Frac(Λ f ) b ⊗ Λ such that for all integers k, j with k ≥ , we have D p (˘ f , ˘ g, α )( k, j ) = D ˘ f ∗ k , Ξ( k, ℓ, α, j ) ord ,p E E ∗ ( f k ) h f k , f k i , where f k and ˘ f k are the eigenforms at level N whose ordinary p -stabilizations are the weight k special-izations of f and ˘ f , and E ∗ ( f k ) := 1 − β p ( f k ) α p ( f k ) − . Combining this with the previous proposition, we have Proposition 5.4.2. For integers k, j with ℓ ≤ j ≤ k − , we have D p (˘ f , ˘ g, α )( k, j ) = E ( f k , g, j ) E ( f k ) · E ∗ ( f k ) · h f k , f k i D ( ˘ f k , ˘ g, α, j ) . Note . We know that the Atkin–Lehner operator gives an isomorphism S ℓ ( N ; K )[ π g ] w N ∼ = ✲ S ℓ ( N ; K )[ π g ∗ ] . Less obviously, there is also an operator S ord ( N ; Λ f )[ π f ] w N ∼ = ✲ S ord ( N ; Λ f )[ π f ∗ ]interpolating the action of the Atkin–Lehner operators on the weight k specializations. To see this, itsuffices to note that the inclusions S k ( N ) ֒ → S k ( N p ) ֒ → S k ( N p ) ֒ → . . . commute with the action of w N and this operator is continuous with respect to the p -adic norm (by the q -expansion principle); theresulting operator on the completion S k ( N p ∞ ) commutes with U p , and hence preserves e ord S k ( N p ∞ ). Proposition 5.4.4. For any k, j ∈ Ω f × Ω , we have D p (˘ f , ˘ g, α )( k, k + ℓ − − j ) = N − j · X y ∈ Z /N Z e πiαx/N D p ( w N ˘ f , w N ˘ g, x/N )( k, j ) . Proof. It suffices to check this result for all pairs of integers k, j with k ≥ j , since these points areZariski-dense in Ω f × Ω. By the classical functional equation, we find that for such k, j we have D p (˘ f , ˘ g, α )( k, k + ℓ − − j ) = A · N − j X y ∈ Z /N Z e πiαx/N D p ( w N ˘ f , w N ˘ g, α )( k, j )where the quantity A is defined by A = E ( f k , g, k + ℓ − − j ) E ( f k ) E ∗ ( f k ) h f k , f k i · (cid:18) E ( f ∗ k , g ∗ , j ) E ( f ∗ k ) E ∗ ( f ∗ k ) h f ∗ k , f ∗ k i (cid:19) − . We obviously have h f ∗ k , f ∗ k i = h f k , f k i . More subtly, we have α p ( f ∗ ) = p k − /β p ( f ) and β p ( f ∗ ) = p k − /α p ( f ); similarly, we have { α p ( g ∗ ) , β p ( g ∗ ) } = { p ℓ − /α p ( g ) , p ℓ − /β p ( g ) } . From these relations, it isclear that E ( f ∗ k ) = E ( f k ), E ∗ ( f ∗ k ) = E ( f k ), and E ( f k , g, k + ℓ − − j ) = E ( f ∗ k , g ∗ , j ). So the ratio A isidentically 1. (cid:3) ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 47 Notation. We write D p ( ˘ f , ˘ g, α ) for the restriction of D p (˘ f , ˘ g, α ) to k × Ω ⊂ Ω f × Ω, where ˘ f is anyelement of S ord ( N ; Λ f )[ π f ] whose specialization in weight k is ˘ f . (This is independent of the choice offamily ˘ f ). Note . If k > ℓ , then the L -function D p ( ˘ f , ˘ g, α ) interpolates the critical values D ( f, g, α, s ); butwhen k = ℓ , there are no such critical values.5.5. The syntomic regulator. Let p be prime and K be a finite extension of Q p with ring of integers O K , and X a smooth proper scheme over O K with generic fibre X . Then there exists a map, the syntomicregulator , r syn : CH ( X , → H ( X/K ) / Fil = (cid:0) Fil H ( X/K ) (cid:1) ∨ , with the property that the diagramCH ( X , ✲ CH ( X, H ( X/K ) / Fil r syn ❄ ⊂ exp ✲ H ( K, H ( X, Q p )(2)) r ´et ❄ commutes (c.f. [Bes00]). Here exp denotes the Bloch–Kato exponential map constructed in [BK90] forthe crystalline G K -representation V = H ( X, Q p )(2). Remark . Note that since X has good reduction, all eigenvalues of Frobenius on D cris ( V ) are Weilnumbers of weight − 2; thus D cris ( V ) ϕ =1 = 0, implying that exp is injective. This also implies that H e ( K, V ) = H f ( K, V ). Note that we do not necessarily have H f ( K, V ) = H g ( K, V ), since V ∗ (1) has allweights equal to 0. It is conjectured that the image of r ´et is precisely H g ( K, V ) but this is only knownin a few special cases, cf. [SS10, Fact 1.1].5.6. Generalization of a theorem of Bertolini–Darmon–Rotger. Note that there is a mapdlog : O ( Y ( N )) × ⊗ Q → M (Γ ( N )) , which corresponds to F ( τ ) F ′ ( τ ) F ( τ ) as functions on H ; and this commutes with the Atkin–Lehnerinvolutions. Proposition 5.6.1. For any α = 0 ∈ Q / Z , we have dlog g ,α = − F (2) α . Proof. Immediate from comparing the q -expansion of F (2) α with that of g ,α , which is given in [Kat04, § (cid:3) We recall the following result, which is a slight reformulation and extension of the main theorem of[BDR12]. Theorem 5.6.2. Let u α be the modular unit on Y ( N ) ⊗ Z ( µ N ) such that dlog u α = e E (2) α , and let ∆ u α be any element of CH ( X ( N ) ⊗ Z ( µ N ) , whose pullback to CH ( Y ( N ) ⊗ Z ( µ N ) , is theclass of (∆ , u α ) where ∆ is the diagonal subvariety.Let f, g be any two newforms of weight 2 and levels N f , N g dividing N , with f ordinary at p , and let ˘ f , ˘ g be test vectors attached to f, g as before. Then we have (cid:16) D p ( ˘ f , ˘ g, α )(2) − D p ( ˘ f , ˘ g, (cid:17) = E ( f, g, E ( f ) · E ∗ ( f ) D r syn (∆ u α ) , pr ∗ ( η ur˘ f ) ∧ pr ∗ ( ω ˘ g ) E . Proof. By Fourier inversion on the multiplicative group ( Z /N Z ) × , which acts on both sides of the claimedformula, it suffices to show that for each Dirichlet character ψ modulo N we have(15) E ( f, g, E ( f ) · E ∗ ( f ) D r syn (∆ u ψ ) , pr ∗ ( η ur˘ f ) ∧ pr ∗ ( ω ˘ g ) E = (P d ∈ ( Z /N Z ) × ψ ( d ) − D p ( ˘ f , ˘ g, dα )(2) if ψ = 1 , P d ∈ ( Z /N Z ) × (cid:16) D p ( ˘ f , ˘ g, dα )(2) − D p ( ˘ f , ˘ g, (cid:17) if ψ = 1 . where u ψ = P d ψ ( d ) − ⊗ u dα ∈ Z ( χ ) ⊗ Z O ( Y ( N ) × ). However, it is clear that both sides of Equation(15) are zero unless ψ = χ := χ − f χ − g , so we may assume ψ = χ .If χ = 1 and α has exact order N , then we can assume without loss of generality that α = 1 /N , andwe are in the case studied in [BDR12]. In the remaining cases, the argument goes through essentiallyidentically. (cid:3) Remark . If in fact χ is primitive modulo N , then both sides are zero unless α has exact order N ,so we may reduce to precisely the case covered by [BDR12].We can now deduce our main theorem of this section. Theorem 5.6.4. Let f, g, ˘ f , ˘ g be as above. Then we have D p ( ˘ f , ˘ g, /N )(1) = − E ( f, g, E ( f ) · E ∗ ( f ) D r syn (Ξ ,N, ) , pr ∗ ( η ur˘ f ) ∧ pr ∗ ( ω ˘ g ) E . Proof. Applying the previous theorem to w N ˘ f and w N ˘ g , we have (cid:16) D p ( w N ˘ f , w N ˘ g, x/N )(2) − D p ( w N ˘ f , w N ˘ g, (cid:17) = E ( f, g, E ( f ) · E ∗ ( f ) D r syn (∆ u x/N ) , pr ∗ ( η ur w N ˘ f ) ∧ pr ∗ ( ω w N ˘ g ) E . We multiply by e πix/N and sum over x ∈ Z /N Z . The left-hand side becomes X x ∈ Z /N Z e πix/N D p ( w N ˘ f , w N ˘ g, x/N )(2) = N D p ( ˘ f , ˘ g, x/N )(1)by the p -adic functional equation.Meanwhile, the right-hand side is X x ∈ x ∈ Z /N Z e πix/N E ( f, g, E ( f ) · E ∗ ( f ) D r syn (∆ u x/N ) , pr ∗ ( η ur w N ˘ f ) ∧ pr ∗ ( ω w N ˘ g ) E . By the functoriality of the syntomic regulator, we have D r syn (∆ u x/N ) , pr ∗ ( η ur w N ˘ f ) ∧ pr ∗ ( ω w N ˘ g ) E = D r syn (∆ ( w ∗ N u x/N ) ) , pr ∗ ( η ur˘ f ) ∧ pr ∗ ( ω ˘ g ) E . As elements of Q ( µ N ) ⊗ Z O ( Y ( N )) × , we have X x ∈ Z /N Z e πix/N ⊗ w ∗ N ( u x/N ) = − N ⊗ g , /N , and the result follows. (cid:3) Remark . One could also prove this statement directly (without the extended detour via Atkin–Lehner involutions and functional equations) by generalizing some of the calculations of [BDR12] to usethe weight 2 Eisenstein series F (2) χ = P x ∈ ( Z /N Z ) × χ ( x ) − F (2) x/N in place of E (2) χ = P x ∈ ( Z /N Z ) × χ ( x ) − E (2) x/N .(Note that F (2) χ is always a holomorphic Eisenstein series if N > 1, while E (2) χ becomes non-holomorphicif χ is the trivial character.) Remark . If we impose slightly more restrictive hypotheses we can avoid the need for any gen-eralization of the main theorem of [BDR12]. If χ is primitive, then it suffices to check that the maintheorem of [BDR12] holds without the assumption that f, g are eigenforms for the U ℓ with ℓ | N ; but thisassumption is not used anywhere in the paper, except in order to explicitly evaluate the Euler factors at ℓ | N . If N f = N g = N then we can dispense with this assumption as well. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 49 Families of cohomology classes In this section, we will construct ´etale cohomology classes from the generalized Beilinson–Flach ele-ments in motivic cohomology defined above, and investigate their properties.6.1. The ´etale regulator. In [Hub00], Huber constructs a p -adic regulator map from motivic cohomol-ogy into Jannsen’s continuous ´etale cohomology: Proposition 6.1.1. Assume that X is a smooth variety over a characteristic 0 field k . Then there is aregulator map (16) r ´et : CH ( X, ✲ H ( X, Z p (2)) Proof. See the second example on [Hub00, p. 772]. (cid:3) Proposition 6.1.2. If X is any smooth variety over k , we have a Hochschild–Serre spectral sequence H p ( k, H q ´et ( X, Z p (2))) ⇒ H p + q cont ( X, Z p (2)) , where H ∗ ( k, − ) denotes continuous Galois cohomology.Proof. See [Jan88a, Remark 3.5]. (cid:3) Corollary 6.1.3. Suppose that X is a smooth affine surface over k . Then we have an edge map (17) H ( X, Z p (2)) → H ( k, H ´et ( X, Z p (2))) . Proof. The fact that X is defined over an algebraically closed field implies that H q ´et ( X, Z p (2)) = 0for q > 2, as a d -dimensional affine variety over an algebraically closed field has ´etale cohomologicaldimension d [Del77, Arcata IV.6.4]. Consequently, we have H ( X, Z p (2)) = 0, and we obtain therequired edge map by Proposition 6.1.2. (cid:3) Corollary 6.1.4. If X is a smooth affine surface over k , the ´etale regulator induces a map (which wealso denote by r ´et by abuse of notation) (18) r ´et : CH ( X, ✲ H ( k, H ´et ( X, Z p (2))) . Proof. Compose r ´et with the edge map (17). (cid:3) The regulator maps have the following functoriality property: Proposition 6.1.5. The regulator maps (18) are compatible with pullback along flat morphisms ofsurfaces X → Y over k , and pushforward along finite morphisms. In particular, they are compatible withthe Galois restriction maps for arbitrary extensions k ′ /k , and with the corestriction maps for finite ones.Proof. This is true essentially by construction for Huber’s regulator into continuous cohomology, since itarises from a realization functor on Voevodsky’s category D M gm of geometrical motives, which in turn isbuilt up from the category (denoted by SmCor in [Hub00]) whose objects are smooth varieties over k andwhose morphisms are finite correspondences X ⇒ Y . It remains only to check that the Hochschild–Serreexact sequence (17) has the required functoriality property, which is standard. (cid:3) The K¨unneth formula. We also recall the K¨unneth formula for ´etale cohomology (cf. [Mil12,Theorem 22.4]): if U and V are varieties of finite type over an algebraically closed field of characteristic0, then we have an exact sequence0 ✲ X r + s = m H r ´et ( U, Z p ) ⊗ Z p H s ´et ( V, Z p ) ✲ H m ´et ( U × V, Z p ) ✲ X r + s = m +1 Tor Z p ( H r ´et ( U, Z p ) , H s ´et ( V, Z p )) ✲ . We are interested in the case when m = 2, and U and V are smooth curves. If U, V are affine, thenthey have ´etale cohomological dimension 1; so the third term vanishes, as do two of the three summandsin the first term, and we have the following result: Lemma 6.2.1. For affine curves U, V , the K¨unneth formula gives an isomorphism H ´et ( U, Z p ) ⊗ Z p H ´et ( V, Z p ) ∼ = ✲ H ´et ( U × V, Z p ) , functorial in U and V and compatible with the Galois action. We shall also need to consider the case when U and V are projective (and connected). In this case, weshall assume the ground field k is Q . By the compatibility of ´etale cohomology with Betti cohomologyafter base extension to C , we find that in this case the ´etale cohomology is Z p in degree 0 or 2, and Z gp in degree 1, where g is the genus. Hence all the Tor terms vanish, since the cohomology groups are free Z p -modules; and we conclude that H ( U × V, Z p ) is the direct sum of H ( U, Z p ) ⊗ Z p H ( V, Z p ) and twoother summands which are both isomorphic (as Galois representations) to Z p ( − Galois representations attached to modular forms. We recall the construction of the Galoisrepresentations attached to cuspidal modular forms of weight 2, using the cohomology of the affinemodular curves Y ( N ). Notation. Let f be a cuspidal modular form of weight 2 and level N . We assume that f is a normalizedeigenform for all the Hecke operators T v (for v ∤ N ) and U v (for v | N ). (We do not assume that f isnew of level N .) As usual, we write a v ( f ) for the v -th Fourier coefficient of f , which is its eigenvaluefor T v if v ∤ N and for U v if v | N ; we also write ε d ( f ) for the eigenvalue of f for the h d i operator for d ∈ ( Z /N Z ) × .By [AS86, Proposition 4], the compactly-supported cohomology H c, Betti ( Y ( N )( C ) , C ) is isomorphicto the space of modular symbols of level Γ ( N ) with coefficients in C . This contains a unique two-dimensional C -linear subspace V C ( f ) on which the Hecke operators T v , U v act as multiplication by theFourier coefficients a v ( f ); and the period isomorphism relating Betti and de Rham cohomology allowsus to regard f as an element of V C ( f ). Moreover, if L is any finite extension of Q containing the Fouriercoefficients of f , V C ( f ) is the base-extension of a two-dimensional L -subspace V L ( f ) ⊆ H ,c ( Y ( N ) , L ).Let p be a prime. Invoking the comparison theorem between (compactly-supported) p -adic and Betticohomology, we can regard Q p ⊗ Q V L ( f ) as a subspace of L ⊗ Q H ,c ( Y ( N ) , Q p ). Both of these are freemodules of rank 2 over L ⊗ Q Q p = Q p | p L p , where the product is over primes of L above p ; so we obtainfor each p a two-dimensional L p -linear subspace V L p ( f ) ⊆ H ,c ( Y ( N ) , L p ).The following proposition is well known: Proposition 6.3.1. The Galois representation V L p ( f ) is “the” irreducible L p -linear Galois representa-tion attached to f . That is, for each prime v ∤ N p , the representation V L p ( f ) is unramified at v and wehave trace L p (cid:0) Frob − v (cid:12)(cid:12) V L p ( f ) (cid:1) = a v ( f ) where Frob v is the arithmetic Frobenius. We note that under Poincar´e duality, the dual space V L ( f ) ∗ is identified with the maximal quotientof H ( Y ( N )( C ) , L ) on which the transposes T ′ v and U ′ v of T v and U v act as multiplication by a v ( f ).Tensoring with Q p , and noting that Poincar´e duality holds in ´etale cohomology with a twist by thecyclotomic character, we obtain an identification of V L p ( f ) ∗ with a quotient of H ( Y ( N ) , L p )(1). Definition 6.3.2. Let O p be the ring of integers of L p . We define T O p ( f ) ∗ as the O p -submodule of V L p ( f ) ∗ generated by the image of H ´et ( Y ( N ) , Z p )(1) , which is a G Q -stable O p -lattice in V L p ( f ) ∗ .Remark . Note that our conventions are somewhat different from those of [Kat04, §§ V L p ( f ) as a subspace of compactly-supported cohomology of a modular curve, while Kato usesthe same symbol to denote a quotient of the non-compactly-supported cohomology. If f is new of level N , then our V O p ( f ) ∗ coincides with the space Kato would denote by V L p ( f )(1) where f is the complexconjugate of N , and similarly for the integral lattices (our T O p ( f ) ∗ is Kato’s V O p ( f )(1)). Remark . One can also define a lattice in V L p ( f ) ∗ using the cohomology of the projective modularcurve. The inclusion Y ( N ) ֒ → X ( N ) induces a pullback map H ( X ( N ) , Z p ) → H ( Y ( N ) , Z p ), whichis injective with cokernel isomorphic to Z r − p where r is the number of cusps. The action of the Heckealgebra on the boundary term Z r − p is Eisenstein, so the map H ( X ( N ) , Q p ) → H ( Y ( N ) , Q p ) is an ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 51 isomorphism on the f -isotypical component. We define e T O p ( f ) ∗ as the image of H ( X ( N ) , Z p ) ⊗ O p in V L p ( f ) ∗ . Note that e T O p ( f ) ∗ ⊆ T O p ( f ) ∗ , and equality holds if f is not congruent modulo p to anEisenstein series.6.4. Generalized Beilinson–Flach classes. Let N ≥ 5. Observe that Y ( N ) ⊗ Q ( µ m ) is a smoothvariety over Q ( µ m ), for any m . By (18), for any prime p we therefore have an ´etale regulator r ´et , Q ( µ m ) : CH ( Y ( N ) ⊗ Q ( µ m ) , ✲ H (cid:16) Q ( µ m ) , H ( Y ( N ) , Z p (2)) (cid:17) . Definition 6.4.1. Let f, g be modular forms of level N which are normalized eigenforms for all theHecke operators T ℓ (for ℓ ∤ N ) and U ℓ (for ℓ | N ), L a number field containing the Fourier coefficientsof f and g , and p a place of L above the rational prime p .Remark . In the situation of definition 6.4.1, we can use the K¨unneth formula (Lemma 6.2.1) toregard T O p ( f, g ) ∗ := T O p ( f ) ∗ ⊗ O p T O p ( g ) ∗ as a quotient of O p ⊗ Z p H ( Y ( N ) , Z p )(2). Definition 6.4.3. Define the map κ f,g, Q ( µ m ) : CH ( Y ( N ) ⊗ Q ( µ m ) , ✲ H ( Q ( µ m ) , T O p ( f, g ) ∗ ) to be the composition of r ´et , Q ( µ m ) with the map on Galois cohomology induced by the projection H ´et ( Y ( N ) , Z p )(2) ✲ T O p ( f, g ) ∗ . Definition 6.4.4. We define the generalized Beilinson–Flach class c z ( f,g,N ) m := κ f,g, Q ( µ m ) ( c Ξ m,N, ) ∈ H ( Q ( µ m ) , T O p ( f, g ) ∗ ) , and its non-integral version z ( f,g,N ) m := κ f,g, Q ( µ m ) (Ξ m,N, ) ∈ H ( Q ( µ m ) , V O p ( f, g ) ∗ ) . The compatibility relations we have shown for the generalized Beilinson–Flach elements for varying m carry over to the cohomology classes: Corollary 6.4.5. For any integers m ≥ , N ≥ , and ℓ a prime such that ℓ | N , we have cores ℓmm (cid:0) c z ( f,g,N ) ℓm (cid:1) = ( ( α f α g ) · c z ( f,g,N ) m if ℓ | m , ( α f α g − σ ℓ ) · c z ( f,g,N ) m if ℓ ∤ m ,where α f , α g are the U ℓ -eigenvalues of f and g , and in the latter case σ ℓ is the arithmetic Frobeniuselement at ℓ in Gal( Q ( µ m ) / Q ) .If ℓ is a prime not dividing mN , then cores ℓmm (cid:0) c z ( f,g,N ) ℓm (cid:1) = σ ℓ (cid:0) ( ℓ − − ε f ( ℓ ) ε g ( ℓ ) σ − ℓ ) − ℓP ℓ ( f, g, ℓ − σ − ℓ ) (cid:1) c z ( f,g,N ) m , where P ℓ ( f, g, X ) is the local Euler factor of f and g at ℓ (cf. Proposition 4.1.2 above).Proof. Immediate from Theorems 3.3.2, 3.4.1 and the compatiblity of the regulator map with corestriction(Proposition 6.1.5). (cid:3) The dependence of c z ( f,g,N ) m on c is as follows: Proposition 6.4.6. There exist classes z ( f,g,N ) m ∈ H ( Q ( µ m ) , V L p ( f, g ) ∗ ) such that the relation (19) c z ( f,g,N ) m = ( c − ε f ( c ) − ε g ( c ) − [ c ] ) z ( f,g,N ) m holds for any c > coprime to mN .Proof. Immediate from Proposition 2.7.5 (5). (cid:3) Proposition 6.4.7. If there exists d ≥ coprime to mN such that d − ε f ( d ) − ε g ( d ) − [ d ] is invertiblein O p [( Z /m Z ) × ] , then there exists z ( f,g,N ) m ∈ H ( Q ( µ m ) , T O p ( f, g ) ∗ ) such that Equation (19) holds in H ( Q ( µ m ) , T O p ( f, g ) ∗ ) (not just modulo torsion).In particular, this holds if the conductor of the reduction modulo p of ε f ε g is divisible by some primewhich does not divide mp .Proof. Clear, since if such a d exists we may define z ( f,g,N ) m := ( d − ε f ( d ) − ε g ( d ) − [ d ] ) − d z ( f,g,N ) m . (cid:3) Local properties of the generalized Beilinson–Flach classes (I). We now study the localproperties of the Beilinson–Flach classes. We shall first recall some standard definitions. Definition 6.5.1. If K is a local field and M is a topological G K -module, we define H nr ( K, M ) to bethe image of the inflation map H ( K nr /K, M I K ) → H ( K, M ) , where I K is the inertia subgroup of G K and K nr the maximal unramifed extension of K .If V is a finite-dimensional Q p -vector space, and ℓ is the residue characteristic of K , we define H f ( K, V ) = ( H nr ( K, V ) if ℓ = p , ker( H ( K, V ) → H ( K, V ⊗ B cris ) if ℓ = p .If T is a Z p -lattice in V stable under G K , we write H f ( K, T ) for the preimage of H f ( K, V ) in H ( K, T ) .(Cf. [BK90] .) Proposition 6.5.2. If T is a finite-rank free Z p -module with a continuous action of G K which is trivialon I K , and ℓ = p , then H f ( K, T ) = H nr ( K, T ) . Proof. We have an inflation-restriction exact sequence0 ✲ H nr ( K, T ) ✲ H ( K, T ) ✲ H ( K nr /K, H ( I K , T )) ✲ , and a corresponding sequence for V in place of T . Suppose x ∈ H f ( K, T ). Then the image of x in H ( I K , V ) is zero, so the image of x in H ( I K , T ) is torsion. However, H ( I K , T ) = Hom( I K , T ) istorsion-free, since T is; thus the image of x in H ( I K , T ) is zero, and hence x ∈ H nr ( K, T ). (cid:3) Definition 6.5.3. If K is a number field and M is a topological G K -module, and v is a prime of K ,we say that x ∈ H ( K, M ) is unramified at v if its image in H ( K v , M ) lies in H nr ( K, M ) . If M is afinite-rank Z p -module or Q p -vector space, and v is a prime above p , we say x is crystalline at v if itsimage in H ( K v , M ) lies in H f ( K v , M ) . Proposition 6.5.4. The generalized Beilinson–Flach class c z ( f,g,N ) m is unramified outside the primesdividing mN p . If p ∤ mN , it is crystalline at the primes above p .Proof. By the preceding proposition, it suffices to check this result after inverting p .Let us choose a prime ℓ ∤ mN p . The compactified modular curve X ( m, mN ) associated to Y ( m, mN )admits a smooth proper model X ( m, mN ) over Z [1 /mN ]; hence it has such a model over Z ℓ . It isclear that the class c Z m,N, lies in the higher Chow group Z ( Y ( m, mN ) , 1) of the integral model of Y ( m, mN ), and we can choose the “negligible elements” of Theorem 2.8.5 in order to obtain a lifting of c Z m,N, to CH ( X ( m, mN ) , ⊗ Q .For proper smooth schemes S over Z ℓ , with ℓ = p , there is a regulator mapCH ( S , ⊗ Q p → H ( S , Q p (2))(see e.g. [Fla92]) compatible with the regulator map r ´et on the generic fibre S . Moreover, the ´etalecohomology H ( S, Q p (2)) is unramified as a representation of G Q ℓ , by the proper base change theo-rem; and the Hochschild–Serre spectral sequence maps H ( S , Q p (2)) to H ( Q nrℓ / Q ℓ , H ( S, Q p (2)) ⊂ H ( Q ℓ , H ( S, Q p (2)), where S = S ⊗ Q ℓ (cf. [Fla92, Lemma 2.3]). Hence the class c z ( f,g,N ) m is unramifiedat the primes above ℓ , as required. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 53 Similarly, if p ∤ mN , we can lift c Z m,N, to a class in CH ( X ( m, mN ) , ⊗ Q where X ( m, mN ) isproper and smooth over Z p . However, the regulator r ´et for proper smooth Z p -schemes takes values in H f , as a consequence of the commutative diagram of § r ´et to the syntomic regulator r syn ; so we are done. (cid:3) Remark . I believe it is known that the regulator map r ´et for arbitrary varieties over p -adic fieldstakes values in H g , as remarked in § p of the Beilinson–Flach classes always lies in this subspace.6.6. Local properties of the generalized Beilinson–Flach classes (II). In order to control thelocal properties of the generalized Beilinson–Flach classes at the “bad” primes, we shall make use of thecompatibility in the p -adic cyclotomic tower, under mild additional hypotheses. Assumption 6.6.1. The level N is divisible by p , and the U p -eigenvalues α f , α g of f and g satisfy v p ( α f α g ) < . Proposition 6.6.2. Suppose Assumption 6.6.1 holds. Then for any m ≥ and any prime v ∤ p of Q ( µ m ) , the cohomology class c z ( f,g,N ) m lies in H f ( Q ( µ m ) v , T O p ( f, g ) ∗ ) .If v p ( α f α g ) = 0 , then it lies in H nr ( Q ( µ m ) v , T O p ( f, g ) ∗ ) .Proof. To lighten the notation, we write K = Q ( µ m ) v , and M = T O p ( f, g ) ∗ . We write K i = Q ( µ mp i ) v (after choosing one of the finitely many primes of Q ( µ mp ∞ ) above v ). Each K i is contained in K nr , since v ∤ p .For each i , there is an inflation-restriction exact sequence0 ✲ H ( K nr /K i , M I v ) ✲ H ( K i , M ) ✲ H ( K nr /K i , H ( I v , M )) ✲ , and the corestriction maps H ( K i +1 , M ) → H ( K i , M ) correspond to the trace maps H ( K nr /K i +1 , H ( I v , M )) → H ( K nr /K i , H ( I v , M )) . Since M is a finitely-generated Z p -module, H ( I v , M ) is finitely generated over Z p , by [Rub00, Propo-sition B.2.7(iii)]; thus the sequence of modules M i = H ( K nr /K i , H ( I v , M )) stabilizes at some i ≫ i ≥ i , the trace maps M i +1 → M i are simply multiplication by p on M i +1 = M i = M ∞ . Let z i be the image of c z ( f,g,N ) mp i in M i . It then follows that for i ≥ i we have( α f α g ) i z = p i − i cores i ( z i ) . If α f α g is a p -adic unit, then this immediately implies that z = 0, since it is divisible by arbitrarily highpowers of p . Thus c z ( f,g,N ) m is unramified at v .Otherwise, we can only deduce that z ∈ (cid:18) p i − i ( α f α g ) i (cid:19) M + ( M ) tors for all i ≫ 0, which implies that z ∈ ( M ) tors as v p ( α f α g ) < 1. Hence the image of z in M ⊗ Q p is zero,so the image of c z ( f,g,N ) m in H ( Q ( µ m ) v , M ⊗ Q p ) is unramified. Thus c z ( f,g,N ) m ∈ H f ( Q ( µ m ) v , M ). (cid:3) Relation between p-stabilized and non-p-stabilized classes. For the arguments of the pre-vious section, we assumed throughout that p | N . If we are given forms of levels prime to p , then we canobtain forms of level divisible by p via “ p -stabilization” (choosing old eigenforms of level divisible by p with the same Hecke eigenvalues at all other primes). In this section, we shall investigate the relationsbetween the classes obtained for the p -stabilized and non- p -stabilized forms.Let f be a normalized eigenform of weight 2 and level N , and let p be a prime such that p ∤ N . Thenthere are two eigenforms f α , f β at level N p in the oldspace attached to f , whose U p -eigenvalues arethe roots α , β of the Hecke polynomial X − a p ( f ) X + pǫ p ( f ). (We assume, by enlarging the field ifnecessary, that these lie in our coefficient field L .) Then there are projection mapspr f α : H ( Y ( N p ) , L p ) → V L p ( f α ) ∗ pr f β : H ( Y ( N p ) , L p ) → V L p ( f β ) ∗ pr f : H ( Y ( N ) , L p ) → V L p ( f ) ∗ and a pushforward map π : H ( Y ( N p ) , L p ) → H ( Y ( N ) , L p ). Proposition 6.7.1. In the above situation, there is a nonzero, G Q -equivariant map π ( α ) : V L p ( f α ) ∗ → V L p ( f ) ∗ and similarly π ( β ) , with the property that π ( α ) ◦ pr f α + π ( β ) ◦ pr f β = pr f ◦ π as maps H ´et ( Y ( N p ) , L p ) → V ∗ f .Proof. Let H ( Y ( N p ) , L p ) [ f ] denote the maximal quotient of H ( Y ( N p ) , L p ) where the operators T ′ v for v ∤ N p and U ′ v for v | N act via a v ( f ). Then, by comparison with modular symbols, we see that H ( Y ( N p ) , L p ) [ f ] is 4-dimensional, and the U ′ p operator on this space is annihilated by the Heckepolynomial.By [CE98, Theorem 2.1], the roots α and β are distinct, so we may write H ( Y ( N p ) , L p ) [ f ] as adirect sum of G Q -stable eigenspaces, which map isomorphically onto the quotients V L p ( f α ) and V L p ( f β ).This gives a lifting of V L p ( f α ) to a subspace of H ( Y ( N p ) , L p ) [ f ] , and the map pr f ◦ π clearly factorsthrough H ( Y ( N p ) , L p ) [ f ] as stated. (cid:3) Now let us suppose we have two normalized weight 2 eigenforms f, g , of level N prime to p as before.Let α, β be the roots of the Hecke polynomial of f at p , and similarly γ, δ for g . By the Coleman–Edixhoven theorem cited above, we have α = β and γ = δ .A choice of root of each polynomial gives p -stabilized eigenforms f α , g γ of level N p . Then for each m we have • a class z ( f,g,N ) m in the cohomology of V L p ( f, g ) ∗ , which is a quotient of H ( Y ( N ) , L p )(2); • an element z ( f α ,g γ ,Np ) m living in the cohomology of the representation V L p ( f α , g γ ) ∗ , which is aquotient of H ( Y ( N p ) , L p )(2).These two representations are isomorphic as abstract Galois representations, but are realized differ-ently as quotients of ´etale cohomology. We can regard both as quotients of the following space: Definition 6.7.2. Let H ´et ( Y ( N p ) , L p ) f,g denote the maximal L p -linear quotient of H ´et ( Y ( N p ) , L p ) on which the operators ( T ′ v , (for v ∤ N p ) and ( U ′ v , (for v | N ) act via the Fourier coefficients of f ,and similarly for g .Note . Using the K¨unneth formula and a modular symbol calculation, we see that H ( Y ( N p ) , L p ) f,g has dimension 16, and can be viewed as a direct sum of four simultaneous eigenspaces for the two opera-tors ( U ′ p , 1) and (1 , U ′ p ), corresponding to the stabilizations ( α, γ ), ( α, δ ), ( β, γ ) and ( β, δ ). Each of theseis a 4-dimensional Gal( Q / Q )-stable L p -linear subspace.For the remainder of this section, we shall assume the following: Assumption 6.7.4. We have αγ = βδ .Remark . Assumption 6.7.4 is a consequence of Assumption 6.6.1, since v p ( αβγδ ) = v p ( p ε f ( p ) ε g ( p )) =2, so if v p ( αγ ) < 1, then v p ( βδ ) > Proposition 6.7.6. If Assumption 6.7.4 is satisfied, then the operator J α,γ := ( U − αδ )( U − βγ )( U − βδ )( αγ − αδ )( αγ − βγ )( αγ − βδ ) , where U = ( U ′ p , U ′ p ) , is an idempotent in End L p H ´et ( Y ( N p ) , L p ) f,g ; it is equal to the identity on the ( α, γ ) eigenspace and zero on the other three eigenspaces. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 55 Proof. We know that αγ = αδ and αγ = βγ by the Coleman–Edixhoven theorem, so if αγ = βδ , the αγ eigenspace for the operator U coincides with the ( α, γ ) simultaneous eigenspace for ( U ′ p , 1) and (1 , U ′ p ).We may thus define a projection onto this eigenspace by applying to U a polynomial that is 1 at αγ andzero at the other three eigenvalues. (cid:3) Proposition 6.7.7. There is a Gal( Q / Q ) -equivariant L p -linear isomorphism π ( α,γ ) : V L p ( f α , g γ ) ∗ ∼ = ✲ V L p ( f, g ) ∗ with the property that π ( α,γ ) ◦ pr ( α,γ ) = π ◦ J α,γ as maps H ´et ( Y ( N p ) , L p ) f,g (2) → V L p ( f, g ) ∗ , where π : H ´et ( Y ( N p ) , L p ) f,g → V L p ( f, g ) ∗ is the natural map induced by the pushforward map Y ( N p ) → Y ( N ) .Proof. We define π ( α,γ ) as π ◦ ι ( α,γ ) , where ι ( α,γ ) is the section of pr α,γ identifying V L p ( f α , g γ ) withthe ( α, γ )-eigenspace of H ( Y ( N p ) , L p ) f,g . The composition ι ( α,γ ) ◦ pr α,γ is therefore equal to theprojection operator J α,γ above, and the proposition follows. (cid:3) Corollary 6.7.8. For p ∤ m we have π ( α,γ ) (cid:16) c z ( f α ,g γ ,Np ) m (cid:17) = αγ (cid:16) − βδp σ − p (cid:17) (cid:16) − αδp σ − p (cid:17) (cid:16) − βγp σ − p (cid:17) ( γ − δ )( α − β ) · c z ( f,g,N ) m . Proof. We shall prove this by a slightly roundabout argument, using the second norm relation “in reverse”to understand how U acts on the zeta elements. Let us write the polynomial( X − αδ )( X − βγ )( X − βδ )( αγ − αδ )( αγ − βγ )( αγ − βδ ) ∈ L p [ X ]as j + j X + j X + j X , and let c z ( f,g,Np ) m be the image of reg ´et ( c Ξ m,Np,j ) in H ( Y ( N p ) , L p ) f,g .Essentially by definition, we have c z ( f α ,g γ ,Np ) m = pr α,γ c z ( f,g,Np ) m , and hence we may apply the precedingproposition to obtain π ( α,γ ) (cid:16) c z ( f α ,g γ ,Np ) m (cid:17) = π (cid:16) J α,γ · c z ( f,g,Np ) m (cid:17) = π (cid:16)(cid:0) j + j U + j U + j U (cid:1) c z ( f α ,g γ ,Np ) m (cid:17) . By the second norm relation for p | N (Theorem 3.3.2) and induction on r , we see that for r ≥ U r (cid:16) c z ( f,g,Np ) m (cid:17) = norm p r mm (cid:16) c z ( f,g,Np ) p r m (cid:17) + σ p · norm p r − mm (cid:16) c z ( f,g,Np ) p r − m (cid:17) + · · · + σ rp · c z ( f,g,Np ) m . On the other hand, by the first norm relation (Theorem 3.1.2) we know that for r ≥ π (cid:16) norm p r mm ( c z ( f,g,Np ) p r m ) (cid:17) = norm p r m m (cid:16) π ( c z ( f,g,Np ) p r m ) (cid:17) = norm p r mm c z f,g,Nm , while for r = 0 we have π (cid:16) c z ( f,g,Np ) m (cid:17) = (1 − ε p ( f ) ε p ( g ) σ − p ) c z ( f,g,N ) m . Combining these statements we have π ( α,γ ) (cid:16) c z ( f α ,g γ ,Np ) m (cid:17) = ( j + j σ p + j σ p + j σ p )(1 − ε p ( f ) ε p ( g ) σ − p ) c z ( f,g,N ) m + ( j + j σ p + j σ p ) norm pmm (cid:16) c z ( f,g,N ) pm (cid:17) + ( j + σ p j ) norm p mm (cid:16) c z ( f,g,N ) p m (cid:17) + j norm p mm (cid:16) c z ( f,g,N ) p m (cid:17) . The prime-to- p case of the second norm relation (Theorem 3.4.1) gives a formula for the second term, andTheorem 3.5.2 extends this to the remaining two terms. Substituting these in, the entirety of the right-hand side simplifies to a linear combination of terms each of which is c z ( f,g,N ) m acted on by a polynomialin σ p , σ − p with coefficients given as rational functions in α, β, γ, δ . After a computation (which was carried out using Sage, [Sage]), one finds the polynomial simplifies to the product of Euler-type factorsstated above. (cid:3) This extremely laborious computation allows us to prove the following theorem, which will be crucialto the Iwasawa-theoretic applications of our Euler system: Corollary 6.7.9. Suppose f, g admit p -stabilizations f α , g γ such that v p ( αγ ) < . Suppose m is coprimeto p and neither of the quantities αδ/p , βγ/p is an r -th root of unity, where r is the order of p in ( Z /m Z ) × .Then for every prime v ∤ p of Q ( µ m ) , the localization of c z f,g,Nm at v lies in H f .Proof. We know from Proposition 6.6.2 above that the class c z f α ,g β ,Npm is in H f at all primes away from p . Since αγ = βδ by Remark 6.7.5, the formula of the previous corollary applies.We note that for λ ∈ L p , the element 1 − λσ − p is invertible in L p [( Z /m Z ) × ] if and only if λ r = 1,where r is the order of σ p as above. It is clear that βδp cannot be a root of unity of any order, as its p -adic valuation is strictly positive. By assumption neither αδ/p nor βγ/p is an r -th root of unity; sothe quantity αγ (cid:16) − βδp σ − p (cid:17) (cid:16) − αδp σ − p (cid:17) (cid:16) − βγp σ − p (cid:17) ( γ − δ )( α − β )is invertible in L p [( Z /m Z ) × ]. Hence c z ( f,g,N ) m is also in H f . (cid:3) Remark . Note that the conclusion of Corollary 6.7.9 does not explicitly mention the choice of p -stabilization ( α, γ ); we use only the fact that one exists. We conjecture that the conclusion holds muchmore generally.6.8. Iwasawa cohomology classes. Notation. We now let S be a finite set of places of Q containing p , ∞ , and all primes whose inertiagroups act nontrivially on T O p ( f, g ) ∗ (which can only happen for primes dividing N ). Let Q S be themaximal extension of Q unramified outside S . Definition 6.8.1. For K a finite extension of Q contained in Q S , i ≥ , and T a topological Z p [ G K ] -module unramified outside S , define H iS ( K, T ) = H i ( Q S /K, T ) . If T is also a finitely-generated Z p -module, and K ∞ is a p -adic Lie extension of K unramified outside S , define H i Iw ,S ( K ∞ , T ) = lim ←− L H iS ( L, T ) where L varies over the set of finite extensions of K contained in K ∞ and the inverse limit is with respectto the corestriction maps.Remark . If K ∞ contains finite extensions of K of degree divisible by arbitrarily large powers of p – for instance, if K ∞ /K is Galois and its Galois group is a p -adic Lie group of positive dimension –then H ,S ( K ∞ , V ) is zero, and H ,S ( K ∞ , T ) is in fact independent of S , as long as S contains the set S consisting of all primes above p or ∞ , all primes ramifying in K ∞ /K and all primes at which T isramified.We now let f, g be eigenforms of level N , with coefficients in a field L , as in Definition 6.4.1. Proposition 6.8.3. Suppose m ≥ and there is no Dirichlet character ψ of conductor dividing mp ∞ such that f ∼ ¯ g ⊗ ψ , where ∼ signifies that these two eigenforms have the same Hecke eigenvalues awayfrom their levels (i.e. correspond to the same newform). Then H ( Q ( µ mp ∞ ) , V L p ( f, g ) ∗ ) = 0 . Proof. The space H ( Q ( µ mp ∞ ) , V L p ( f, g ) ∗ is preserved by the residual action of the abelian groupGal( Q ( µ mp ∞ ) / Q ), so if it is nonzero, it contains a subspace on which Gal( Q ( µ mp ∞ ) / Q ) acts by somecharacter λ (possibly after a finite extension of the field L ). This gives a nonzero Gal( Q / Q )-equivarianthomomorphism V L p ( f ) → V L p ( g ) ∗ ( λ ) = V L p (¯ g )( χλ ) ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 57 where χ is the cyclotomic character. Since both sides are irreducible representations of Gal( Q / Q ), thismap must be an isomorphism; consequently ψ = χλ has finite order, and f ∼ ¯ g ⊗ ψ . (cid:3) We can now prove the main result of this section, which shows that the elements ( α f α g ) − ic z ( f,g,N ) mp i for i ≥ unbounded ) Euler system: Theorem 6.8.4. Let m, N ≥ with ( m, p ) = 1 , and let p be a prime dividing N . Let f, g be modularforms of level N which are eigenforms for all the Hecke operators, with U p -eigenvalues α f and α g suchthat h := v p ( α f α g ) < . Suppose that f ¯ g ⊗ ψ for all Dirichlet characters ψ of conductor dividing mp ∞ . Then for any r such that h ≤ r < , there is a unique element c z ( f,g,N ) m,r ∈ H r (Γ) ⊗ Λ(Γ) H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ ) whose projection to H S ( Q ( µ mp i ) , V L p ( f, g ) ∗ ) is equal to ( α f α g ) − ic z ( f,g,N ) mp i if i ≥ , and to (1 − ( α f α g ) − σ p ) c z ( f,g,N ) m if i = 0 .Moreover, if ℓ is a prime not dividing mN , the corestriction map sends c z ( f,g,N ) ℓm,r to σ ℓ (cid:0) ( ℓ − − ε f ( ℓ ) ε g ( ℓ ) σ − ℓ ) − ℓP ℓ ( ℓ − σ − ℓ ) (cid:1) c z ( f,g,N ) m,r . Proof. The existence of c z ( f,g,N ) m,r satisfying the projection formula for i ≥ i = 0 follows from the i = 0 case of Corollary6.4.5. (cid:3) Note . The elements c z ( f,g,N ) m,r are in fact independent of r ∈ [ h, v p ( α f α g ) ≤ r < r ′ < 1, then c z ( f,g,N ) m,r ′ is the image of c z ( f,g,N ) m,r under the natural map H r (Γ) ⊗ Λ(Γ) H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ ) ✲ H r ′ (Γ) ⊗ Λ(Γ) H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ )induced by the inclusion H r (Γ) ֒ → H r ′ (Γ).In the case when v p ( α f α g ) = 0, we can prove a stronger result; in this case we can dispense with theassumption that f is not a twist of ¯ g , and we even get integral coefficients. Theorem 6.8.6. Assume that f and g are eigenforms of level dividing N , and such that v p ( α f α g ) = 0 .Then there is a unique element c z ( f,g,N ) m ∈ H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ ) whose projection to H S ( Q ( µ mp i ) , T O p ( f, g ) ∗ ) is equal to ( ( α f α g ) − ic z ( f,g,N ) mp i if i ≥ (cid:0) − ( α f α g ) − σ p (cid:1) c z ( f,g,N ) m if i = 0 .Proof. For i ≥ 1, let c z ( f,g,N ) m,i = ( α f α g ) − ic z ( f,g,N ) mp i . As v p ( α f α g ) = 0, we have c z ( f,g,N ) m,i ∈ H S ( Q ( µ mp i ) , T O p ( f, g ) ∗ ). Moreover, it is clear from Corollary 6.4.5that cores i/i − ( c z ( f,g,N ) m,i ) = c z ( f,g,N ) m,i − , i.e. c z ( f,g,N ) m = (cid:0) c z ( f,g,N ) m,i (cid:1) i ≥ defines an element in H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ ). The projection formulafor i = 0 follows as before. (cid:3) Remark . The difficulties arising when v p ( α f α g ) ≥ f ′ , g ′ of level prime to p , at most two of the four possible choices of p -stablizations f, g of f ′ , g ′ will bepossible, and in many cases (e.g. if a p ( f ′ ) = a p ( g ′ ) = 0) there are no valid choices at all.Our methods using higher Chow groups can perhaps be thought of as an “algebraic avatar” of themodular symbol computations of [AV75]. It is interesting to speculate whether the overconvergentmodular symbols of [PS11] also admit such an algebraic analogue, which could conceivably be appliedin the critical-slope cases. Dispensing with c . We now investigate the extent to which the “smoothing factor” c may beremoved. Notation. If R is a integral domain, we write Q ( R ) for its field of fractions. For integers c and m suchthat ( c, m ) = 1, we write [ c ] for the image of σ c in Z p [Gal( Q ( µ m ) / Q )].Under the hypotheses of Theorem 6.8.6, there exists an element z ( f,g,N ) m ∈ H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ ) ⊗ Λ(Γ ) Q (Λ(Γ ))such that(20) c z ( f,g,N ) m = ( c − ε f ( c ) − ε g ( c ) − [ c ] ) z ( f,g,N ) m . Remark . Note that we are identifying Γ with Gal( Q ( µ mp ∞ ) / Q ( µ mp )). We may define z ( f,g,N ) m as( d − ε f ( d ) − ε g ( d ) − [ d ] ) − d z ( f,g,N ) m for any d > N and congruent to 1 mod mp ; then [ d ] lies in Γ , so the expression iswell-defined, and it is evidently independent of the choice of d . Notation. Write Γ ( m ) = Gal( Q ( µ mp ∞ ) / Q ) ∼ = ( Z /m Z ) × × Γ. Lemma 6.8.9. Let p be a prime ideal of Λ(Γ ( m ) ) of height which does not contain p . If the conductorof the Dirichlet character ε f ε g does not divide mp ∞ , then there exists an integer c > coprime to mpN such that c − ε f ( c ) − ε g ( c ) − [ c ] / ∈ p .Proof. Since p does not contain p , it corresponds to a Galois orbit of continuous characters Γ ( m ) → Q p .Let κ p be a representative of this orbit. Define ˜ h : Γ ( m ) → Q × p by ˜ h ( x ) = κ p ( x ) /χ ( x ) where χ : Γ ( m ) → Z × p is the p -adic cyclotomic character. We need to show that there is an integer c ≥ mpN such that ˜ h ([ c ]) = ε f ( c ) ε g ( c ).However, if no such integer existed, then ε f ε g would have to factor through the natural map b Z × → Γ ( m ) , i.e. would have to have conductor dividing mp ∞ , contrary to our hypotheses. (cid:3) Corollary 6.8.10. If ε f ε g does not have conductor dividing mp ∞ , then z ( f,g,N ) m ∈ H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ ) ⊗ Q . Proof. Let Z be the Λ(Γ ( m ) )-module generated by c z ( f,g,N ) m for all possible c and let Z be the Λ(Γ ( m ) )-module generated by z ( f,g,N ) m . By (20), Z ⊂ Z and there exists µ ∈ Λ(Γ ( m ) ) such that µZ ⊂ Z and Z /µZ is p -torsion free. Hence, it is enough to show that Z , p = Z , p for any prime ideal p of height 1which does not contain p . Fix such a p . By Lemma 6.8.9, there exists c such that c − ε f ( c ) − ε g ( c ) − [ c ] / ∈ p , so z ( f,g,N ) m ∈ Z , p by (20), as required. (cid:3) Remark . Note that if the mod p reduction of the Dirichlet character ε f ε g does not have conductordividing mp ∞ , then we can even deduce that z ( f,g,N ) m ∈ H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ ) . Lemma 6.8.12. If the residual representation of T O p ( f, g ) ∗ restricted to G Q µm is irreducible, then H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ ) is a free Λ(Γ ( m ) ) -module.Proof. This follows from the argument of [Kat04, § T = T O p ( f, g ) ∗ , Λ = Λ(Γ ( m ) ) and H ( T ) = H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ ) for simplicity. It is enough toshow that if ( x, y ) is a maximal ideal of Λ, the two maps α : H ( T ) x ✲ H ( T ) and β : H ( T ) /xH ( T ) y ✲ H ( T ) /xH ( T )are injective.If x = p , the injectivity of α follows from the fact that lim ←− n H ( Q ( µ mp n ) , T /p ) = 0, which is aconsequence of the finiteness of T /p . If x is such that Λ /x Λ is p -torsion free, it is enough to showthat H ( Q ( µ m ) , T ⊗ Λ /x Λ) = 0, where the action of σ ∈ G Q ( µ m ) on Λ is given by multiplication by ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 59 ¯ σ − , where ¯ σ denotes the image of σ in Γ ( m ) . But T is irreducible, so non-abelian. This implies that H ( Q ( µ m ) , T ⊗ Λ /x Λ) = 0.To show that β is injective, it is enough to show that H ( Z [1 /mp ] , T ⊗ Λ / ( x, y )) = 0. But Λ / ( x, y ) ∼ = O p / M p ( r ) for some r , so we are done by the irreducibility of T / M p . (cid:3) Corollary 6.8.13. If ε f ε g does not have p -power conductor and the residual representation of T O p ( f, g ) ∗ restricted to G Q µm is irreducible, then z ( f,g,N ) m ∈ H ,S ( Q ( µ mp ∞ ) , T O p ( f, g ) ∗ ) . Proof. This follows from the argument in [Kat04, § Z be the Λ(Γ ( m ) )-module generated by z ( f,g,N ) m . By the proof of Corollary 6.8.10, Z , p ⊂ H ,S ( Q ( µ mp ∞ ) p for any prime ideal p of Λ(Γ ( m ) )height one. But H ,S ( Q ( µ mp ∞ ) is a free Λ(Γ ( m ) )-module by Lemma 6.8.12, hence the result. (cid:3) Variation in Hida families. We now make use of the first norm relation (Theorem 3.1.2) to buildelements in the cohomology of towers of modular curves, under additional ordinarity hypotheses. Let usbegin by recalling some of Ohta’s results in [Oht99, Oht00] concerning the structure of the module GES p ( N, Z p ) := lim ←− n ≥ H ( Y ( N p n ) , Z p ) , which can be roughly summarized by the statement that one can build a Hida theory for this moduleafter replacing the usual Hecke operators with their transposes.More precisely, we note that the full Hecke algebra does not act on GES p ( N, Z p ), since the operator U p = (cid:18) p (cid:19) does not commute with the trace maps. Rather, we obtain an action of the Hecke operator U ′ p , corresponding to (cid:18) p 00 1 (cid:19) . Ohta shows that one may use the operator U ′ p to define an “anti-ordinaryprojector” e ′ ord = lim n →∞ ( U ′ p ) n ! , analogous to the usual Hida ordinary projector e ord = lim n →∞ ( U p ) n ! .Ohta proves the following control theorem for the anti-ordinary part of GES p ( N ) Z p , which is naturallya module over the Iwasawa algebra of Γ = (1 + p Z p ) × via the diamond operators: Proposition 6.9.1 ([Oht99, 1.3, 1.4]) . The module e ′ ord GES p ( N ) Z p is free of finite rank over Λ(Γ ) ,and for each r ≥ there is a G Q -equivariant isomorphism e ′ ord GES p ( N, Z p ) /ω r ∼ = e ′ ord H ( Y ( N p r ) , Z p ) , where ω r is the kernel of the natural map Λ(Γ ) → Z p [( Z /p r Z ) × ] . From this isomorphism, we deduce that e ′ ord GES p ( N, Z p ) has an action of the Hecke algebrae ′ ord H ′ ( N, Z p ) = lim ←− r ≥ e ′ ord H ′ (Γ ( N p r ) , Z p ) , where H ′ (Γ ( N p r ) , Z p ) is the Z p -subalgebra of End Q p M (Γ ( N p r ) , Q p ) generated by the Hecke operators T ′ ( n ) for n ≥ h q i for q ∈ ( Z /N Z ) × . Here T ′ ( ℓ ), for ℓ prime, corresponds to the double coset (cid:18) ℓ 00 1 (cid:19) , so in particular for ( m, N p ) = 1 we have T ′ ( m ) = h m i − T ( m ) (the adjoint of T ( m ) withrespect to the Petersson product).The algebra e ′ ord H ′ ( N, Z p ) algebra is finite and free as a Λ(Γ )-module. (Note that it is not generallyfree as a Λ(Γ)-module, although it is evidently projective). Proposition 6.9.2 ([Oht99, 2.2]) . The Hecke algebra e ′ ord H ′ ( N, Z p ) is isomorphic to the Hecke algebra H ord ( N, Z p ) := e ord H ( N, Z p ) acting on the module e ord M ( N, Λ(Γ )) of ordinary Λ -adic modular forms(not necessarily cuspidal), via the map sending T ( n ) ′ to T ( n ) and h n i to h n i − . Definition 6.9.3. In the above situation, by a Hida family of tame level N , we mean a maximal idealof the ring H ord ( N, Z p ) . For each Hida family g , we define T ( g ) ∗ = (e ′ ord GES p ( N, Z p )) g (1) . Corollary 6.9.4. Let g be an ordinary weight 2 Hecke eigenform of level N p s , with coefficients in somefinite extension L p / Q p with ring of integers O p . Then we have an isomorphism of O p -linear Galoisrepresentations O p ⊗ H ord ( N, Z p ) T ( g ) ∗ ∼ = T O p ( g ) ∗ , where T O p ( g ) ∗ is the representation defined in 6.3 above.Proof. Clear from the definition of T O p ( g ) ∗ and the control theorem (Theorem 6.9.1). (cid:3) Theorem 6.9.5. Let N ≥ be prime to p . If g is a Hida family of tame level N , and f is any eigenformof level N p k for k ≥ whose U p -eigenvalue α f satisfies v p ( α f ) < , then for each integer m ≥ there isa cohomology class c z ( f, g ) m ∈ H S ( Q ( µ m ) , T O p ( f ) ∗ ⊗ Z p T ( g ) ∗ ) , such that for each classical weight 2 specialization g of g with coefficients in L , the image of c z ( f, g ) m in H ( Q ( µ m ) , T O p ( f ) ∗ ⊗ O p T O p ( g ) ∗ ) = H ( Q ( µ m ) , T O p ( f, g ) ∗ ) is the generalized Beilinson–Flach element c z ( f,g,N ′ ) m , where N ′ is the greatest common divisor of thelevels of f and g .Proof. We know that the elements c Ξ m,Np s , for s ≥ S , and are compatibleunder pushforward via the natural projection maps. Hence the sequence of elements defined by pushingforward c Ξ m,Np s , to CH ( Y ( N p r ) × Y ( N p s ) × Q ( µ m ) , s ≥ r , are compatible under pushforwardmaps in the Y ( N p s ) factor alone. Applying the ´etale regulator, we obtain elements of the modulelim ←− s ≥ r H ( Q ( µ m ) , H ( Y ( N p r ) × Y ( N p s ) , Z p )(2)) . For each s , using the K¨unneth formula we may decompose H ( Y ( N p r ) × Y ( N p s ) , Z p ) as the tensorproduct of the H ’s of the two factors. Projecting to the quotient T O p ( f ) of H ( Y ( N p r ) , Z p )(1), andapplying the anti-ordinary projector e ′ ord to H ( Y ( N p s ) , Z p )(1), we may argue exactly as in Proposition6.6.2 above to deduce that the elements we obtain are unramified outside N p .Since the restricted-ramification cohomology groups H iS ( Q ( µ m ) , − ) commute with inverse limits, weobtain an element of H S ( Q ( µ m ) , T O p ( f ) ∗ ⊗ e ′ ord GES p ( N ) Z p (1)) . Pushing forward along the canonical map e ′ ord GES p ( N ) Z p (1) → T ( g ) ∗ , we obtain the required elements. (cid:3) We also obtain a corresponding result for the product of two Hida families, whose proof is essentiallyidentical to the above: Theorem 6.9.6. Let N ≥ be prime to p . If f , g are Hida families of tame level N , then for eachinteger m ≥ there is a cohomology class c z ( f , g ) m ∈ H S ( Q ( µ m ) , T ( f ) ∗ ˆ ⊗ Z p T ( g ) ∗ ) , such that for classical weight 2 specializations f , g of f , g with coefficients in L , the image of c z ( f , g ) m in H ( Q ( µ m ) , T O p ( f ) ∗ ⊗ O p T O p ( g ) ∗ ) = H ( Q ( µ m ) , T O p ( f, g ) ∗ ) is the generalized Beilinson–Flach element c z ( f,g,N ′ ) m , where N ′ is the greatest common divisor of thelevels of f and g .Remark . We do not know if one can formulate a result analogous to Theorem 6.8.4 incorporatingHida-family variation in g , since we do not know whether the results of Appendix A.2 apply for “big”Galois representations; but if f is ordinary there are no such issues. Theorem 6.9.8. In the situation of Theorem 6.9.6, for each m prime to p there exists a cohomologyclass c z ( f , g ) m ∈ H ,S ( Q ( µ mp ∞ ) , T ( f ) ∗ ˆ ⊗ Z p T ( g ) ∗ ) whose image in H ,S ( Q ( µ mp i ) , T ( f ) ∗ ˆ ⊗ Z p T ( g ) ∗ ) for each i ≥ is equal to ( α f α g ) − i · c z ( f , g ) mp i . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 61 Integrality of the Poincar´e pairing. In this section, we prove a technical lemma that will beneeded in our applications to bounding Selmer groups. We assume that p > N ≥ 5, and p ∤ N .Recall that X ( N ) admits a canonical smooth proper model over Z [1 /N ] ([DR73]) and hence over Z p .By [FM87], the integral de Rham cohomology H ( X ( N ) , Ω • X ( N ) / Z p ) is a filtered Dieudonn´e moduleover Z p , and T ( H ( X ( N ) , Ω • X ( N ) / Z p )) = H ( X ( N ) , Z p ) , where T ( − ) is the Fontaine–Laffaille functor.We define versions of these in the f -isotypical component by projection. As in Remark 6.3.4 above,we define e T O p ( f ) ∗ to be the image of H ( X ( N ) , Z p ) ⊗ O p in V L p ( f ) ∗ , and similarly for g . We define D cris ( e T O p ( f ) ∗ ) as the image of H ( X ( N ) , Ω • X ( N ) / Z p ) ⊗ Z p O p in D cris ( V L p ( f ) ∗ ); then D cris ( e T O p ( f ) ∗ ) isa strongly divisible O p -lattice, and its image under T ( − ) is e T O p ( f ) ∗ .Let us recall here the definition of η ur f . Definition 6.10.1. Let X = X ( N ) and η ah f = ¯ f ∗ ( z ) d ¯ z h f ∗ , f ∗ i k,N ∈ H ( X C ) . We denote η f its image in H ( X/ C , O X/ C ) , which lies in H ( X/ Q p , O X/ Q p ) ; then η ur f is defined to bethe lift of η f to the unit root subspace of H ( X C p ) f, ur . Our aim is to investigate the denominator of the class η ur f relative to the sublattice O L ⊗ Z p H ( X ( N ) , Ω • X ( N ) / Z p ) ⊆ L p ⊗ Q p H ( X ( N ) , Ω • X ( N ) / Q p ) . Since the unit root lifting is obviously integral, it suffices to show that η f ∈ O L ⊗ Z p H ( X ( N ) , O X ( N ) / Z p ). Proposition 6.10.2. An element of L p ⊗ H ( X ( N ) / Q p , O X ( N ) / Q p ) lies in the sublattice O p ⊗ H ( X ( N ) , O X ( N ) / Z p ) if and only if it pairs to an element of O p with all elements of O p ⊗ H ( X ( N ) / Z p , Ω X ( N ) / Z p ) .Proof. Since the pairing between H ( X ( N ) , O X ( N ) / Z p ) and H ( X ( N ) / Z p , Ω X ( N ) / Z p ) is defined over Z p , it suffices to assume O p = Z p . But Serre duality shows that this pairing is perfect, i.e. identifies H ( X ( N ) / Z p , Ω X ( N ) / Z p ) with the Z p -dual of H ( X ( N ) / Z p , O X ( N ) / Z p ). (cid:3) Lemma 6.10.3. Let φ ∈ S ( N ; L p ) . Then the element ω φ of H ( X/L, Ω X/L ) lies in H ( X/ O L , Ω X/ O L ) if and only if φ ∈ S ( N ; O p ) .Proof. We have by definition ω φ ( q ) = φ ( q ) dq/q , which is defined over O L if φ is. (cid:3) Definition 6.10.4. If f ∈ S (Γ ( N ) , L ) , let I f denote the ideal in O such that (cid:26) h f ∗ , φ ih f ∗ , f ∗ i : φ ∈ S ( N, O ) (cid:27) = I − f . Remark . Note that I − f contains O , so I f is an integral ideal (rather than a fractional ideal). Theideal I f essentially measures the extent to which f is congruent to other eigenforms in S ( N, O ). Corollary 6.10.6. For any prime p ∤ N , we have η f ∈ I − f · O p ⊗ Z p H ( X ( N ) , O X ( N ) / Z p ) . Proof. By the construction of the class η f , for any ϕ ∈ S (Γ ( N ) , O ) we have h η f , ω φ i = h f ∗ , φ ih f ∗ , f ∗ i ∈ I − f O , so the result follows by Lemma 6.10.3 and Proposition 6.10.2. (cid:3) Corollary 6.10.7. The linear functional D cris ( V L p ( f, g ) ∗ ) → L p given by pairing with η ur f ⊗ ω g maps the submodule D cris ( e T O p ( f ) ∗ ) ⊗ D cris ( e T O p ( g ) ∗ ) into I − f O p . Proposition 6.10.8. Let z ∈ H f (cid:16) Q p , h e T O p ( f ) ⊗ e T O p ( g ) i ∗ (cid:17) . Then h log( z ) , η ur f ⊗ ω g i ∈ I − f · (1 − α − γ − ) − (1 − α − δ − ) − · O p , where α is the unit root of the Hecke polynomial of f and β, δ are the roots of the Hecke polynomial of g .Proof. By Fontaine–Laffaille theory, for any crystalline O p -linear G Q p -representation V whose Hodgefiltration has length < p and such that D cris ( V ) ϕ =1 = 0, the maplog Q p ,V : H f ( Q p , V ) ∼ = ✲ D cris ( V )Fil D cris ( V )induces an isomorphism of O p -modules H f ( Q p , T )torsion ∼ = ✲ (1 − ϕ ) − D (1 − ϕ ) − D ∩ Fil D cris ( V )for any G Q p -stable lattice T ⊆ V with corresponding strongly divisible lattice D ⊆ D cris ( V ); cf. Theorem4.1 and Lemma 4.5 of [BK90].In our case we may take V = W ⊗ V L p ( g ) ∗ where W is the 1-dimensional unramified quotient of V L p ( f ) ∗ , since the linear functional given by pairing with η ur f ⊗ ω g factors through this quotient. Let ussuppose that g is ordinary; using the explicit description of the strongly divisible lattices in D cris ( V L p ( g ) ∗ )given in [LZ12b, § 5] one checks that(1 − ϕ ) − D (1 − ϕ ) − D ∩ Fil D cris ( V ) ⊆ p − k · DD ∩ Fil D cris ( V ) , where k = v p (cid:2) (1 − α − γ − )(1 − α − δ − ) (cid:3) . In the non-ordinary case one reasons similarly using thedescription of the Wach module of the (unique up to scaling) lattice in D cris ( V L p ( g ) ∗ ) given in [BLZ04].Combining this with Corollary 6.10.7 gives the result. (cid:3) Bounding strict Selmer groups Let f , g be newforms of weight 2, level N and characters χ f and χ g , respectively. Let L be the subfieldof Q generated by the coefficients of f and g . For a prime p of L , denote by V L p ( f ) and V L p ( g ) the L p -representations of G Q attached to f and g , respectively. The aim of this section is to apply Theorem7.1.5 below to the representation V L p ( f ) ⊗ V L p ( g ).7.1. The method of Euler systems. We recall some definitions and results from [Rub00]. Let O be thering of integers of a finite extension E/ Q p , and let T be a free O -module of finite rank with a continuousaction of G Q which is unramified at almost all primes. Let V = T ⊗ O E and W = V /T = T ⊗ O E/ O .Let Σ be a finite set of primes containing p and all prime numbers at which the action of G Q on T ramifies. Let A be a set of integers such that • if m ∈ A , then all divisors of m are in A ; • if r, s ∈ A , then LCM ( r, s ) ∈ A ; • ℓ ∈ A for all primes ℓ / ∈ Σ.For a prime ℓ Σ, define p ℓ ( X ) = det E (cid:0) − Frob − ℓ X (cid:12)(cid:12) V ∗ (1) (cid:1) ∈ Z p [ X ] , where Frob ℓ is the arithmetic Frobenius at ℓ . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 63 Definition 7.1.1 (C.f. [Rub00, Definition 2.1.1]) . An Euler system for ( T, A, Σ) is a system of elements c m ∈ H ( Q ( µ m ) , T ) for all m ∈ A, such that if ℓ is a prime such that m, mℓ ∈ A and ℓ = Σ , then the corestriction map H ( Q ( µ ℓm ) , T ) → H ( Q ( µ m ) , T ) sends c ℓm to ( p ℓ ( σ − ℓ ) c m if ℓ ∤ m and ℓ = Σ , c m if ℓ | m or ℓ ∈ Σ .Here, σ ℓ denotes the arithmetic Frobenius of ℓ in Gal( Q ( µ m ) / Q ) .Remark . Our notations differ slightly from those of [Rub00]. Firstly, Rubin writes T ∗ for the“Tate dual” Hom( T, Z p (1)), while we write this as T ∗ (1). More significantly, Rubin considers an infiniteabelian extension K and a class c F for every finite subextension F of K ; in our case K is the extension K A = Q ( µ r : r ∈ A ), and it suffices to specify a class for each subextension of the form Q ( µ m ), which isour c m , and to fill in the remainder via corestriction. Definition 7.1.3. For each prime ℓ , let H f ( Q ℓ , W ) be the image of H f ( Q ℓ , V ) in H ( Q ℓ , W ) .Define S { p } ( Q , W ) = ker (cid:0) H ( Q , W ) ✲ M ℓ = p H ( Q ℓ , W ) /H f ( Q ℓ , W ) (cid:1) , and define the strict Selmer group of W over Q as S { p } ( Q , W ) = ker (cid:0) S { p } ( Q , W ) ✲ H ( Q p , W ) (cid:1) . (Thus S { p } is the Selmer group with local conditions given by the Bloch–Kato condition at primesaway from p and the zero local condition at p .)We define S { p } ( Q , T ) similarly, and also S { p } ( K, T ) similarly, for any number field K . (We shall onlyneed this when K = Q ( µ m ), see Hypothesis Hyp( S ( p ) , V ) below.)In order to state the main theorem, we introduce the following sets of hypotheses. Note that Hyp( Q , T )is strictly stronger than Hyp( Q , V ), but Hyp( p, A ) and Hyp( S { p } , V ) are independent of each other. Hypothesis (Hyp( Q , T )) . T ⊗ k is an irreducible k [ G Q ]-module, where k is the residue field of O ; andthere exists an element τ ∈ G Q which satisfies the following conditions:(i) τ acts trivially on µ p ∞ ;(ii) T / ( τ − T is free of rank 1 over O . Hypothesis (Hyp( Q , V )) . V is an irreducible E [ G Q ]-module; and there exists an element τ ∈ G Q whichsatisfies the following conditions:(i) τ acts trivially on µ p ∞ ;(ii) dim Q p ( V / ( τ − V ) = 1. Hypothesis (Hyp( p, A )) . The set A contains all powers of p . Hypothesis (Hyp( S { p } , V )) . The following three conditions hold:(i) T G Q = 0;(ii) c m ∈ S { p } ( Q ( µ m ) , T ) for all m ∈ A ;(iii) there exists an element γ ∈ G Q such that • γ acts trivially on µ p ∞ , • γ − T . Theorem 7.1.4. Assume that V is not the trivial representation, and that Hypothesis Hyp( Q , V ) andat least one of hypotheses Hyp( p ) and Hyp( S ( p ) , V ) are satisfied. If c = ( c m ) m ∈ A is an Euler systemfor ( T, A, Σ) , and the image of c in H ( Q , T ) is not contained in H ( Q , T ) tors , then S { p } ( Q , W ∗ (1)) isfinite. Theorem 7.1.5. Assume that p > and that Hypothesis Hyp( Q , T ) and at least one of hypotheses Hyp( p ) and Hyp( S ( p ) , V ) are satisfied. If c = ( c m ) m ∈ A is an Euler system for ( T, A, Σ) , then length O ( S { p } ( Q , W ∗ (1))) ≤ ind O ( c ) + n W + n ∗ W where ind O ( c ) is the largest power of the maximal ideal by which c can be divided in H ( Q , T ) / torsion,and the quantities n W and n ∗ W are as defined in Theorem 2.2.2 of [Rub00] .Proofs. If Hyp( p, A ) holds, then Theorem 7.1.4 and Theorem 7.1.5 are Theorem 2.2.3 and Theorem 2.2.2of [Rub00] respectively. If instead Hyp( S ( p ) , V ) holds, then the necessary modifications to the proofs areoutlined in § op.cit. . (cid:3) Verifying the hypotheses on T. The main result of this section is Proposition 7.2.18 below,which implies that under some mild technical assumptions there is a large supply of primes where thecondition Hyp( Q , T ) is satisfied.7.2.1. Big image results for one modular form. We begin by some results from [Mom81] and [Rib85]regarding the image of the Galois representations attached to a modular form. Let f = P n ≥ a n q n bea new eigenform of weight k ≥ 2, level N and character ǫ , not of CM type. Let L = Q ( a n : n ≥ 1) be itscoefficient field, with ring of integers O L .Recall that an extra twist of f is an element γ ∈ Gal( L/ Q ) such that γ ( f ) is equal to the twist of f by some Dirichlet character χ γ . We let Γ ⊆ Gal( L/ Q ) be the group of such γ , and F ⊆ L the fixed fieldof Γ; and we let H ⊆ Gal( Q / Q ) be the absolute Galois group of the finite abelian extension K cut outby the Dirichlet characters χ γ .For each prime λ of L , it is clear that the trace of the Galois representation ρ L λ ( f ) | H takes values in F µ , where µ is the prime of F below λ . Theorem 7.2.1 (Momose–Ribet; see [Rib85, Theorem 3.1]) . For all but finitely many λ , the image ofthe Galois representation ρ L λ ( f ) | H is a conjugate of the group { g ∈ GL ( O F,µ ) : det( g ) ∈ Z × ℓ } where µ and ℓ are the primes of F and Q below λ .Remark . For a “generic” modular form f , there will be no extra twists if the character f is trivial,but there will always be at least one if f has nontrivial character, since the complex conjugate f ∗ is atwist of f .We will need the following slight strengthening: Proposition 7.2.3. Let K ′ be any finite extension of K which is abelian over Q , and let H ′ ⊆ H beits absolute Galois group. Then for all but finitely many λ , the image of ρ L λ ( f ) | H ′ is a conjugate of thegroup { g ∈ GL ( O F,µ ) : det( g ) ∈ Z × ℓ } above.Proof. If λ is a prime satisfying the conclusion of the theorem, then the image of ρ L λ ( f ) | H ′ containsSL ( O F,µ ), since SL ( O F,µ ) is equal to its own commutator subgroup. But for all but finitely manyprimes ℓ , the field K ′ is linearly disjoint from Q ( µ ℓ ∞ ) and thus the cyclotomic character is a surjection H ′ → Z ℓ . (cid:3) Big image results for pairs of modular forms. We recall the following result from group theory: Proposition 7.2.4 (Goursat’s Lemma, cf. [Lan02, Exercise I.5]) . Let G , G be groups and H a subgroupof G = G × G such that the projections π i : H → G i are surjective. Let N = H ∩ ( G × { e } ) and N = H ∩ ( { e } × G ) , which we identify with subgroups of G , G in the obvious manner. Then the N i are normal in G i , and H is the graph of an isomorphism G /N ∼ = G /N . Corollary 7.2.5. Let F , F ′ be finite fields of the same characteristic, both of order ≥ . Let H be asubgroup of SL ( F ) × SL ( F ′ ) surjecting onto both factors. Then either H is the whole of SL ( F ) × SL ( F ′ ) ,or F = F ′ and H is conjugate in GL ( F ) × GL ( F ) to one of the following subgroups:(i) the diagonal subgroup { ( x, ϕ j ( x )) : x ∈ G } , for some ≤ j < k , where F = F p k and ϕ is the p -power Frobenius of F ;(ii) the subgroup { ( x, y ) : y = ± ϕ j ( x ) } , for some ≤ j < k .Proof. This follows immediately from Goursat’s lemma and a case-by-case check, given that the groupsPSL ( F ) for fields F of order ≥ ( F ) and PSL ( F ) are both isomorphic to PGL ( F ) ⋊ h ϕ i . (cid:3) ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 65 Proposition 7.2.6. Let O , O ′ be the rings of integers of any two unramified extensions of Q p , where p is a prime ≥ , with residue fields F , F ′ . Then any closed subgroup H ⊆ SL ( O ) × SL ( O ′ ) which surjectsonto SL ( F ) × SL ( F ′ ) must be the whole of SL ( O ) × SL ( O ′ ) .Proof. We follow the argument given for SL ( Z p ) by Swinnerton-Dyer in [SD73]. It suffices to show thatfor each n ≥ 2, the image H n of H in SL ( O /p n ) × SL ( O ′ /p n ) contains the subgroups K n × × K ′ n , where K n is the kernel of SL ( O /p n ) → SL ( O /p n − ) and similarly for K ′ n . Note that for each n , the group K n is abelian, and is isomorphic (via m p n − m ) to the group of trace zero matricesin M ( F ), which is generated by matrices u such that u = 0.We now proceed by induction on n . Let u ∈ M ( F ) satisfy u = 0. By assumption, we may then find h ∈ H congruent to (1 + u, 1) modulo p ; and, as shown in op.cit. , we have h p = (1 + pu, 1) mod p . Thus(1 + pu, ∈ H , and thus H ⊇ K × 1. Similarly, H contains 1 × K ′ , so in fact H is the whole ofSL ( O /p ) × SL ( O ′ /p ).Suppose n ≥ H n − is everything. We claim H n contains K n × 1. Again, K n consists ofmatrices of the form (1 + p n − u, h ∈ H congruent to(1 + p n − u, 1) modulo p n − . Then h p is congruent to (1 + p n − u, 1) modulo p n , so (1 + p n − u, ∈ H n .Thus H n ⊇ K n × H n ⊇ × K ′ n , so we are done. (cid:3) As a corollary, we obtain the following result. Proposition 7.2.7. Let O , O ′ be as above, with characteristic ≥ , and let H be a subgroup of SL ( O ) × SL ( O ′ ) which surjects onto both factors. Then either H = SL ( O ) × SL ( O ′ ) , or O ′ = O and H iscontained in the subgroup { ( x, y ) ∈ SL ( O ) : x = ± ϕ j y mod p } for some j . We shall now boost this to a statement about GL . For O , O ′ as before, let G denote the group { ( x, y ) ∈ GL ( O ) × GL ( O ′ ) : det( x ) = det( y ) ∈ Z × p } . We can regard this as a fibre product G × Z × p G , where G = { x ∈ GL ( O ) : det( x ) ∈ Z × p } and similarlyfor G . Proposition 7.2.8. Let H be a subgroup of G which surjects onto G and G . Then either H = G , orwe have O = O ′ and H is contained in the subgroup of G given by { ( x, y ) : x = ± ϕ j y mod p } for some j .Proof. Let G ◦ = SL ( O ) × SL ( O ′ ) and let H ◦ = H ∩ G ◦ . Then H ◦ has full image in each of SL ( O )and SL ( O ′ ), so either H ◦ = G ◦ , or O ′ = O and the image of H ◦ modulo p is contained in the subgroup { ( x, y ) : x = ± ϕ j y mod p } for some j .Suppose first that H ◦ = G ◦ . Then we must have H = G , since for each g ∈ G , there is some h ∈ H with det( h ) = det( g ), and then h − g lies in G ◦ so by assumption it must be in H .In the remaining case, by replacing H with its image under the automorphism ϕ j × 1, we may assumewithout loss of generality that j = 0. Then any h ∈ H ◦ is of the form ( x, y ) with x = ± y modulo p . Let( x, y ) be any element of H , and consider the class of t = x − y in PSL ( F ); then for any ( u, v ) ∈ H ◦ , wehave [ u − tu ] = [ u − x − yu ] = [ x − ][( xux − ) − ( yvy − )][ y ][ v − u ] = [ x − y ] = [ t ] , since ( xux − , yvy − ) ∈ H ◦ . Thus the classes [ t ] and [ u ] commute in PSL ( F ). However, since H ◦ surjectsonto SL ( O ), this forces [ t ] to be in the centre of PSL ( F ), which is trivial (since it is a simple group).Thus x = ± y mod p for all ( x, y ) ∈ H , as claimed. (cid:3) Assume now that we have two newforms f and g , and let L be the subfield of Q generated by thecoefficients of f and g . For each prime p of L , we may consider the image of the Galois representation ρ f, p × ρ g, p : Gal( Q / Q ) → GL ( L p ) × GL ( L p ).Let H be the subgroup of Gal( Q / Q ) cut out by the Dirichlet characters corresponding to the “extratwists” of f and g , and let K be its fixed field (an abelian extension of Q ). Let F, F ′ be the subfieldsof L fixed by the extra twists. By Proposition 7.2.3, we know that for all but finitely many p , theimage of ρ f, p | H is the group { x ∈ GL ( O ) : det( x ) ∈ Z × p } , where O is the completion of F at theprime below p and p is the residue characteristic of p ; similarly, the image of ρ g, p | H will be the group { x ∈ GL ( O ′ ) : det( x ) ∈ Z × p } where O ′ is the completion of F ′ at p . Then the image of the Galois representation ρ f, p × ρ g, p is a subgroup of the group G p = G defined above, which surjects onto eitherfactor. Proposition 7.2.9. In the above situation, either the image of H under ρ f, p × ρ g, p is G p , or O ′ = O and there is an element γ ∈ Gal( L/ Q ) and a quadratic character χ : H → {± } such that the equality (21) ρ f, p ( σ ) = ± ρ f, p ( σ ) γ mod p holds for all σ ∈ H .Proof. We know from above that if the image of ρ f, p × ρ g, p is not G , then O = O ′ and ρ f, p ( σ ) = ± ϕ j ρ g, p ( σ ) mod p for all σ ∈ H , where ϕ j is the mod p Frobenius.Now we may take γ to be any element of the decomposition group of p in Gal( L/ Q ) reducing to ϕ j modulo p . (Of course, there will almost always be only be one such element, since only finitely manyprimes ramify in L/ Q .) (cid:3) We now lift to characteristic 0. Let w be a prime of the field K ; we define a w ( f ) = tr ρ f,λ ( σ − w ),where σ w is the arithmetic Frobenius at w in H and λ is some prime of L ; if w is a degree 1 prime, thenthis is just a v ( f ) where v is the rational prime below w , and for general primes w it may be expressedas a polynomial in a v ( f ) and χ v ( f ). In any case it is obviously independent of the choice of auxilliaryprime λ , and (since K is abelian over Q ) it depends only on the prime v of Q below w . We define a w ( g )similarly. Definition 7.2.10. Let us say a prime p of residue characteristic ≥ is a good prime for the pair ( f, g ) if the image of H under ρ f, p × ρ g, p is the whole of G p . If the image is a proper subgroup, but has fullprojection to either factor, we say p is a bad prime .Remark . If p divides 2 or 3, or is such that ρ f, p or ρ g, p has small image, we consider p to beneutral, neither good nor bad. By the theorem of Momose–Ribet (Theorem 7.2.1), there are only finitelymany neutral primes. Corollary 7.2.12. If there are infinitely many bad primes for ( f, g ) , then there is γ ∈ Gal( L/ Q ) suchthat the equality a w ( f ) = ± γ ( a w ( g )) for all primes w of K .Proof. For each bad prime p , there exists a γ ∈ Gal( L/ Q ) such that the congruence (21) holds, and inparticular (by taking σ = σ − w ) we have a w ( f ) = γ ( a w ( g ) ) mod p for all primes w of K .Since Gal( L/ Q ) is finite, there exists some γ such that the congruence (21) of the proposition holdsfor all p in an infinite set B . In this case, we have a w ( f ) = γ ( a w ( g ) ) mod p for infinitely many p . So we must have an equality a w ( f ) = γ ( a w ( g ) ), since a nonzero element of anumber field cannot be divisible by infinitely many primes. (cid:3) Corollary 7.2.13. If there are infinitely many bad primes for ( f, g ) , there exists a quadratic Groessen-character κ of K (equivalently, a continuous quadratic character of H ) such that a w ( f ) = κ ( w ) a w ( g ) for all primes w of K .Proof. This follows from the strong multiplicity one theorem for SL /K , cf. [Ram00]: the Satake pa-rameters of the base-change representations BC ( π f ) and BC ( π γg ) of GL ( A K ) agree up to sign at anyprime w , and Ramakrishnan’s result guarantees that the sign relating the two is given by a quadraticcharacter. (cid:3) Remark . Frustratingly it does not seem to be possible to show the existence of κ without suchheavy automorphic machinery, even though we know that for infinitely many primes p the sign relating ρ f and ρ g modulo p is given by a character. Of course, we can define the quantity a w ( f ) intrinsically in “automorphic” terms, as (up to normalizations) it is thetrace of the d -th power of the conjugacy class in GL ( C ) which is the Satake parameter of π f,v , where d is the degree ofthe unramified extension [ K w : Q v ]; this makes the independence of λ selfevident. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 67 Theorem 7.2.15. Suppose there are infinitely many bad primes for ( f, g ) . Then f is Galois-conjugateto some twist of g .Proof. From Ramakrishan’s theorem, we know that we have ρ f,λ | H ∼ = ρ γg,λ | H ⊗ τ for a quadratic character τ of H , and any choice of prime λ . Inducing up from H to H ′ = Gal( Q / Q ),we have ρ f,λ ⊗ Ind H ′ H (1 H ) = ρ g,λ ⊗ Ind H ′ H ( τ ) . But the left-hand side contains ρ f,λ as a direct summand, while the right-hand side is a direct sum ofrepresentations of the form ρ γg,λ ⊗ µ where µ is an irreducible Artin representation. Hence there mustbe at least one µ which is one-dimensional and such that ρ f,λ ∼ = ρ γg,λ ⊗ µ , in which case we must have f = g γ ⊗ µ . (cid:3) Existence of the special element. As in the previous section, let f and g be two newforms, and let L be the subfield of Q generated by the coefficients of f and g . We assume that f is not Galois conjugateto a twist of g , so by Theorem 7.2.15 there are only finitely many bad primes for ( f, g ). We retain thenotation of the previous section.Let p be a good prime which does not divide the levels of f and g , and p the rational prime below p .We make the following crucial assumption: Assumption 7.2.16. The character χ = ( χ f χ g ) − is nontrivial, and its conductor is not a power of p . For simplification, we also make the following assumption: Assumption 7.2.17. We have L f, p = L g, p = Q p , so after a suitable choice of basis, we may assumethat the image of ρ f, p × ρ g, p is contained in GL ( Z p ) × GL ( Z p ) . We can now prove the main result of this section. Proposition 7.2.18. There exists an element τ ∈ G Q ( µ p ∞ ) such that if V / ( τ − V is 1-dimensional,where V = V L p ( f, g ) ∗ .If χ is not congruent modulo p to any character of p -power conductor, then there exists τ such that T / ( τ − T is free of rank 1, for any G Q -stable lattice T in V .Proof. Choose some α ∈ Gal( Q / Q ) such that χ ( α ) = 1, but α is in the kernel of the p -adic cyclotomiccharacter. Note that such α do exist, since the conductor of χ not a power of p . Consider the coset α · ( H ∩ G Q ( µ p ∞ ) ). Since p is a good prime, under ρ f, p × ρ g, p , the coset α · ( H ∩ G Q ( µ p ∞ ) ) is mapped to( ρ f, p ( α ) , ρ g, p ( α )) · SL ( Z p ) , which consists of all pairs ( u, v ) of matrices such that det( u ) = χ f ( α ) and det( v ) = χ g ( α ). In particular,it contains the pair (cid:18) x x − χ f ( α ) (cid:19) (cid:18) x − xχ g ( α ) (cid:19) for any x ∈ Z × p . The image of this pair under the tensor product homomorphism GL × GL → GL isthe diagonal matrix with entries (cid:2) , x − χ f ( α ) , x χ g ( α ) , χ f ( α ) χ g ( α ) (cid:3) . By choosing x appropriately, we can arrange that neither x − χ f ( α ) nor x χ g ( α ) is equal to 1. Thus 1 isan eigenvalue of τ on V L p ( f ) ⊗ V L p ( g ) with multiplicitly exactly 1.If we assume the stronger condition on χ in the statement, then we can assume that χ f ( α ) χ g ( α )is not 1 modulo p . By choosing x appropriately we can assume that x − χ f ( α ) and x χ g ( α ) are alsonon-congruent to 1, so it follows that T / ( τ − T is free of rank 1 as required (for any τ -stable O L -latticein V , and in particular any G Q -stable lattice). (cid:3) The quantities n W and n ∗ W . We recall the definitions of the quantities n W and n ∗ W in Rubin’stheory. Let T be a finite-rank free O -module with a continuous action of G Q . As usual, write V = T ⊗ E and W = V /T . Definition 7.2.19. Define Ω to be the smallest extension of Q whose Galois group acts trivially on W and on µ p ∞ , and define n W = ℓ O (cid:16) H (Ω / Q , W ) ∩ S { p } ( K, W ) (cid:17) n ∗ W = ℓ O (cid:0) H (Ω / Q , W ∗ (1)) ∩ S { p } ( K, W ∗ (1)) (cid:1) . We now give conditions under which these quantities are zero. Proposition 7.2.20. Suppose the centre of Ω acts on each of T ⊗ k and T ∗ (1) ⊗ k via a nontrivialcharacter. Then n W = n ∗ W = 0 .Proof. We shall show that the hypotheses imply that H (Ω / Q , W ) = H (Ω / Q , W ∗ (1)) = 0. We give theargument for W ; the proof for W ∗ (1) is similar.Clearly we have H (Ω / Q , V ) = 0, and hence H (Ω / Q , W ) is finite. So it suffices to show that H (Ω / Q , W )[ ̟ ] = 0 where ̟ is a uniformizer. But we have a surjection H (Ω / Q , W [ ̟ ]) ։ H (Ω / Q , W )[ ̟ ],so we are reduced to showing that H (Ω / Q , W [ ̟ ]) = H (Ω / Q , T ⊗ k ) is zero. However, this is immediatesince any representation of nontrivial central character cannot have a nontrivial extension by the trivialrepresentation. (cid:3) The Euler system. As above, let f and g be newforms of weight 2, level N and characters χ f and χ g , respectively. Let p be a prime not dividing N . Let L be a number field containing the coefficientsof f α and g β , and let p be a prime of L above p . Let E = L p and O its ring of integers. We write T = T O ( f, g ) ∗ and p ℓ ( X ) = det(1 − Frob − q X | T ∗ (1)) = P ℓ ( f, g, ℓ − X ) ∈ O L [ X ]. We assume that thefollowing conditions are satisfied: Assumption 7.3.1. (i) the character χ = χ f χ g is not trivial, and moreover is not trivial modulo p ;(ii) there exist p -stabilizations f α , g γ of f and g with U p -eigenvalues α, γ respectively such that • v p ( αγ ) < , • α/γ is not a root of unity. Fix c ≥ N , and let A be the set of square-free integers prime to N pc . By Corollary 6.4.5,we have, for every integer m ∈ A , a cohomology class c z f,g,Nm ∈ H ( Q m , T ) which satisfy the followingcompatibility property: if m ∈ A and ℓ is a prime comprime to mN pc , then the image of z f,g,Nℓm underthe corestriction map H ( Q ℓm , T ) → H ( Q m , T ) is − σ ℓ A ℓ ( σ − ℓ ) z f,g,Nm , where A ℓ ( X ) is a polynomial in O L [ X ] congruent modulo ℓ − p ℓ ( X ). Lemma 7.3.2. There exists a system of cohomology classes { c ˜ z f,g,Nm ∈ H ( Q m , T ) : m ∈ A } such that c ˜ z f,g,N = c z f,g,N and if m ∈ A and ℓ is a prime such that mℓ ∈ A , then the image of c ˜ z f,g,Nℓm under the corestriction map H ( Q ℓm , T ) → H ( Q m , T ) is A ℓ ( σ − ℓ ) c ˜ z f,g,Nm . Proof. By induction on the number of prime factors of m , we can choose (non-canonically) a system ofelements γ m ∈ ( Z /m Z ) × for every m ∈ A such that γ mℓ = ℓ − γ m mod m . Identify γ m with an elementof Gal( Q ( µ m / Q ) via the inverse of the cyclotomic character, and define c ˜ z f,g,Nm = ( − s ( m ) γ m · c z f,g,Nm ,where s ( m ) is the number of prime factors of m . It is clear by construction that the elements have therequired property. (cid:3) Note . By Corollary 6.7.9, the classes c z f,g,Nm are in the Selmer group S { p } ( Q ( µ m ) , T O p ( f, g ) ∗ ). As S { p } ( Q ( µ m ) , T O p ( f, g ) ∗ ) is invariant under the action of Gal( Q ( µ m ) / Q ), it follows that the same is truefor the modified classes c ˜ z f,g,Nm . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 69 We now show that we can convert the classes ( c z f,g,Nm ) m ∈ A into an Euler system. Let T , Σ and A be as defined at the beginning of Section 7.1. Then we have the following result (c.f. [Rub00, Lemma9.6.1]): Lemma 7.3.4. Suppose that for all primes ℓ ∈ A we have polynomials r ℓ ( X ) , s ℓ ( X ) ∈ O [ X ] such that r ℓ ( X ) ≡ s ℓ ( X ) (mod ℓ − , and suppose that we have a collection of cohomology classes (cid:8) ˜ c m ∈ H ( Q ( µ m ) , T ) : m ∈ A (cid:9) such that if ℓ ∈ A is coprime to m , then cores Q ( µ ℓm ) / Q ( µ m ) (˜ c ℓm ) = ( r ℓ ( σ − ℓ )˜ c m if ℓ ∤ m ˜ c m if ℓ | m . Then there exists a collection of classes (cid:8) c m ∈ H ( Q ( µ m ) , T ) : m ∈ A (cid:9) with the following properties:(i) For all m , c m ∈ O [( Z /m Z ) × ] · ˜ c m . (ii) if ℓ is a prime such that m, mℓ ∈ A , then cores Q ( µ ℓm ) / Q ( µ m ) ( c ℓm ) = ( s ℓ ( σ − ℓ ) c m if ℓ ∤ mc m if ℓ | m (iii) if m ∈ A and χ is a character of Gal( Q ( µ m ) / Q ) of conductor k such that prime( m ) ⊂ prime( k ) ∪ Σ ,then X γ ∈ Gal( Q ( µ m ) / Q ) χ ( γ ) γ ( c m ) = X γ ∈ Gal( Q ( µ m ) / Q ) χ ( γ ) γ (˜ c m ) . Definition 7.3.5. Define { c ˆ z f,g,Nm ∈ H ( Q ( µ m ) , T O p ( f, g ) ∗ )) : m ∈ A } to be the classes obtained byapplying Lemma 7.3.4 to our classes c ˜ z f,g,Nm , where we take r ℓ ( X ) = A ℓ ( X ) and s ℓ ( X ) = p ℓ ( X ) = P ℓ ( f, g, ℓ − X ) , and as above A is the set of square-free integers coprime to N pc .Note . By construction, the classes { c ˆ z f,g,Nm : m ∈ A } are an Euler system for ( T, Σ , A ) in thesense of Definition 7.1.1, where Σ is the set of primes dividing N pc . Moreover, because of (i), we have c ˆ z m ∈ S { p } ( Q ( µ m ) , T ) for all m .7.4. Finiteness of the strict Selmer group. We now combine the above results to prove a finitenesstheorem for the strict Selmer group. For the convenience of the reader, we shall recapitulate all of theassumptions we have made on f and g . Assumption 7.4.1. Assume that f and g are weight 2 newforms with coefficients in a number field L ,and p a prime of L above the rational prime p , with the following properties:(i) Neither f nor g is of CM type.(ii) f is not a twist of g .(iii) The character ε f ε g is non-trivial.(iv) p ≥ .(v) p does not divide the levels of f and g .(vi) p is totally split in the field L , so L p = Q p .(vii) The p -adic Galois representations of f and g are surjective onto GL ( Z p ) .(viii) There exists some prime v such that χ ( v ) = 1 for all inner twists χ of f or g , and a v ( f ) = ± a v ( g ) mod p .(ix) f is ordinary at p .(x) There exists a root γ of the Hecke polynomial of g at p such that v p ( γ ) < and α/γ is not a rootof unity, where α is the unit root of the Hecke polynomial of f . If we assume hypotheses (i)–(iii) (which do not depend on p ), then there will be many p such that theremaining hypotheses hold. Theorem 7.4.2. Suppose Assumption 7.4.1 is satisfied, and the p -adic Rankin–Selberg L -function D p ( f, g, /N ) does not vanish at , where N is some integer divisible by the levels of f and g . Then S { p } (cid:18) Q , V L p ( f, g ) T O p ( f, g ) (1) (cid:19) < ∞ . Proof. It suffices to show that the hypotheses of Theorem 7.1.4 are satisfied for T = T O p ( f, g ) ∗ . ByProposition 7.2.18, the element τ required by Hypothesis Hyp( Q , V ) exists; and the Euler system ofDefinition 7.3.5 satisfies Hypothesis Hyp( S { p } , V )(ii). Since T is nontrivial and irreducible, T G K = 0;and the element γ in Hypothesis Hyp( S { p } , V )(iii) clearly exists.By Theorem 5.6.4, if D p ( f, g, /N )(1) = 0, the image of reg p Ξ ,N, in the ( f, g )-isotypical quotient of H ( X ( N ) / Q p ) / Fil is nonzero. Hence, by the diagram of § z f,g,N at p is nonzero, so in particular c z f,g,N is non-torsion as an element of H ( Q , T L p ( f, g ) ∗ )for any c > 1. Thus we may apply Theorem 7.1.4 to the Euler system ( c ˆ z f,g,Nm ) m ∈ A of Definition 7.3.5to obtain the finiteness of the strict Selmer group. (cid:3) The order of the strict Selmer group. Theorem 7.1.5 gives a bound for the order of the strictSelmer group, under slightly stronger hypotheses than Theorem 7.4.2. Theorem 7.5.1. Suppose Assumption 7.4.1 is satisfied, and in addition the mod p reduction of ε f ε g isnot trivial. Then we have length Z p S { p } (cid:18) Q , V L p ( f, g ) T O p ( f, g ) (1) (cid:19) ≤ v p (cid:18) (1 − p − βα )(1 − p − βγ )(1 − p − βδ ) D p ( f, g, /N )(1) (cid:19) + λ where λ is the p -adic valuation of the ideal I f of Definition 6.10.4 above.Proof. Our condition on the mod p reduction of ε f ε g implies that Hypothesis Hyp( Q , T ) is satisfied(again by Proposition 7.2.18; note that the mod p reduction cannot be a nontrivial character of p -powerconductor as p does not divide the levels of f and g ). The condition also assures that the quantities n W and n ∗ W appearing in Theorem 7.1.5 are zero (Proposition 7.2.20).We consider the linear functional α on H ( Q , V L p ( f, g ) ∗ ) given by x 7→ h log Q p ( x ) , η ur f ⊗ ω g i . On thelattice e T O p ( f, g ) ∗ this takes values in I − f · (1 − α − γ − ) − (1 − α − δ − ) − O p , by Corollary 6.10.7; but sincethe Galois representations of f and g are assumed to have big image, we have e T O p ( f, g ) ∗ = T O p ( f, g ) ∗ .Theorem 5.6.4 shows that τ maps the class z f,g, to E ( f ) E ∗ ( f ) E ( f, g, D p ( f, g, /N )(1) . Hence the index of divisibility of z f,g,N is bounded above by v p (cid:18) E ( f ) E ∗ ( f )(1 − α − γ − )(1 − α − δ − ) E ( f, g, D p ( f, g, /N )(1) (cid:19) + λ. We can ignore the factor E ∗ ( f ) := 1 − βα − , since α f is a unit and β f is a non-unit so E ∗ ( f ) ∈ O × p .Substituting the definitions of E ( f ) and E ( f, g, E ( f )(1 − α − γ − )(1 − α − δ − ) E ( f, g, 1) = (1 − p − βα )(1 − p − βγ )(1 − p − βδ ) . (cid:3) An example. It may seem slightly unclear whether the long list of conditions in Assumption 7.4.1may be simultaenously satisfied, so we present the following explicit example (computed using Sage[Sage]).Let f be the unique weight 2 newform of level 11 (corresponding to the elliptic curve E : y + y = x − x ); and let g be the unique newform of weight 2, level 26, and character χ := (cid:0) • (cid:1) with a ( g ) = i ,so the q -expansions of f and g are f = q − q − q + 2 q + q + O ( q ) ,g = q + iq − q − q − iq + O ( q ) . Note that χ has conductor 13, so the local component of π g at 2 is an unramified twist of the Steinbergrepresentation; on the other hand f is unramified principal series at 2 and Steinberg at 11. So f cannotbe a twist of g , and neither f nor g is of CM type (since CM forms cannot be Steinberg at any prime).The form f has no inner twists (since it is non-CM and has coefficients in Q ); as for g , its Galois orbitconsists of g and ¯ g , so its only inner twist is ¯ g . ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 71 To calculate the image of the Galois representations of f and g , we note that Sage [Sage] has a facilityto compute all the exceptional primes for the Galois representation attached to an elliptic curve (i.e. thoseprimes for which the image of the Galois representation is not GL ( Z p )). This speedily tells us that ρ f,p is surjective for all p = 5.The form g does not correspond to an elliptic curve, but there is a Dirichlet character ψ : ( Z / Z ) × → Q ( i ) × such that g ⊗ ψ corresponds to an elliptic curve E ′ of conductor 2 × = 338 (the curve withCremona label 338d, given by y + xy = x + x + 504 x − E ′ are { , } . Letting H = G Q ( √ , the kernel of the character χ , we see that for all primes p / ∈ { , , , } the image of H under ρ g, p , for any prime p of Q ( i ) above p , is GL ( Z p ).Moreover, the only prime such that a v ( f ) = a v ( g ) for all v split in Q ( √ 13) is p = 5. We deduce thatfor any p congruent to 1 mod 4 and not in { , } , and any prime p of Q ( i ) above p , the hypotheses (i)– (viii) are satisfied.We check that both f and g are ordinary at the primes above 17 (it doesn’t matter which prime wetake, since a ( g ) ∈ Z ); and for any choice of roots α, γ of roots of the Hecke polynomials of f and g ,the minimal polynomial of α/γ over Q is x + x − x + x + 1, so in particular α/γ is not a rootof unity. Thus hypotheses (ix) and (x) are satisfied if p is either of the primes above 17.8. Conjectures on higher-rank Euler systems We now explain how the cohomology classes constructed in the previous section may be reconciled withthe general conjectural setup of cyclotomic Iwasawa theory for motivic Galois representations formulatedby Perrin-Riou, and its extension to the two-variable situation as formulated by the second and thirdauthors in [LZ12a].8.1. Euler systems: rank 1 and higher rank. Let us place ourselves again in the general setting of § T is a free O -module with a continuous action of G Q unramified outside a finite set Σ ∋ p ,and A is a set of integers satisfying the conditions loc.cit. . Suppose that all integers in A are coprime to p . Perrin-Riou’s conjectures, as formulated in [PR98] (cf. also [Rub00, § Conjecture 8.1.1. An Euler–Iwasawa system of rank r ≥ consists of the data of, for each m ∈ A , aclass c m ∈ r ^ Λ(Γ m ) H ( Q ( µ mp ∞ ) , V ) with the property that if ℓ is prime and ℓ, mℓ ∈ A , we have cores Q ( µ mℓp ∞ ) Q ( µ mp ∞ ) c mℓ = ( p ℓ ( σ − ℓ ) c m if ℓ ∤ m Σ , c m if ℓ | m Σ . Note that a rank 1 Euler–Iwasawa system is equivalent to the data of an Euler system for ( T, A p , Σ)in the previous sense, where A p = { p k m : m ∈ A, k ≥ } .As noted in [PR98, § r − 1” elements. We make the following definition: Definition 8.1.2. We define a Perrin-Riou functional to be the data of, for each squarefree m prime to S as above, an element Φ m ∈ r − ^ Hom Λ (cid:0) H ,S ( Q ( µ mp ∞ ) , T ) , Λ (cid:1) ι , with the property that for each ℓ ∤ mS , we have Φ m = Φ mℓ ◦ res Q ( µ mℓp ∞ ) Q ( µ mp ∞ ) . Lemma 1.2.3 of op.cit. shows that if ( c p ( m )) is an Euler system of rank r , and (Φ m ) is a Perrin-Rioufunctional, then the elements Φ m ( c p ( m )) ∈ H ,S ( Q ( µ mp ∞ ) , T ) define an Euler system of rank 1. (Here, as explained loc.cit., we interpret Φ m as a map r ^ H ,S ( Q ( µ mp ∞ ) , T ) → H ,S ( Q ( µ mp ∞ ) , T )which we also denote by Φ m .)Appendix B of op.cit. shows that (under mild hypotheses on T ) there is a plentiful supply of Perrin-Riou functionals, although there is no obvious canonical choice. More specifically, given any m and anyΦ m , there exists a Perrin-Riou functional extending Φ m . Hence, given as a starting point a rank r Eulersystem, one may construct a rank 1 Euler system (indeed many such systems) and obtain Iwasawa-theoretic results from this rank 1 system; but these rank 1 Euler systems are noncanonical, and inparticular there is no reason to expect that they should have any relation to L -values. Remark . An alternative approach to bounding Selmer groups in the r > Otsuki’s functionals. We now explain a construction due to Otsuki [Ots09], who has shown howto construct canonical linear functionals on cohomology groups by composing the dual exponential mapwith an appropriate “weighted trace”. These maps do not satisfy the compatibility properties of aPerrin-Riou functional, and thus give rise to systems of elements of group rings satisfying a modifiedcompatibility property; we shall show that this modification is consistent with the results we have shownfor our generalized Beilinson–Flach classes.For technical reasons we shall work in the limit over the cyclotomic extension, rather than directlyover Q ( µ m ); this avoids problems caused by zeroes of local Euler factors (cf. the discussion at the startof § ζ m ∈ Q for all m ≥ ζ nmn = ζ m for all integers m, n .Let G m = Gal( Q ( µ m ) / Q ) and Γ m = Gal( Q ( µ mp ∞ ) / Q ); we identify Γ m with G m × Γ in the obvious way.Let V be an E -linear p -adic representation of G Q , where E/ Q p is a finite extension, which is crystallineat p with non-negative Hodge–Tate weights and such that no eigenvalue of Frobenius on D cris ( V ) is aroot of unity. Then for all m ≥ p -adic regulator map L Γ Q ( µ m ) ,V : H ( Q ( µ mp ∞ ) , V ) ✲ Q ( µ m ) ⊗ Q H E (Γ) ⊗ E D cris ( V )is well-defined (as the sum of the local regulator maps at the primes of Q ( µ m ) above p ).Let D = D cris ( V ∗ ) = D cris ( V ) ∗ , and let D mp ∞ = Λ E (Γ) ⊗ E D ⊗ Q Q ( µ m ). We regard D mp ∞ as aΓ m -module, via the usual action of G m on Q ( µ m ) and of Γ on Λ E (Γ). Following [Kur02] and [Ots09],we make the following definition: Definition 8.2.1. Define a pairing t m : D mp ∞ × H ( Q ( µ mp ∞ ) , V ) ✲ H E [Γ m ] by t m ( x, z ) = X σ ∈ G m [ σ ] trace Q ( µ m ) / Q D σx, L Γ Q ( µ m ) ,V ( z ) E cris . Here we extend h , i cris to be Γ-linear in the second variable and Γ-antilinear in the first. One checksthat t m ( σx, τ z ) = [ σ − τ ] · t m ( x, z )for all σ, τ ∈ Γ m (not just in Γ).Now fix two families F ℓ , G ℓ of polynomials in E [ X ], indexed by primes ℓ / ∈ Σ, such that F ℓ , G ℓ ∈ XE [ X ] for all ℓ .Let A be the set of square-free integers prime to Σ. For each prime ℓ / ∈ Σ and each m ∈ A , considerthe Λ(Γ m )-linear endomorphism of Λ E (Γ) ⊗ Q Q ( µ m ) given by ˆ σ ℓ ( x ⊗ ζ ) = τ ℓ x ⊗ ζ ℓ , for all roots of unity ζ ∈ µ m , where τ ℓ is the arithmetic Frobenius at ℓ in Γ. Thus ˆ σ ℓ is the action of the Frobenius at ℓ in Γ m if ℓ ∤ m , and is a possibly non-invertible endomorphism if ℓ | m ; and the ˆ σ ℓ all commute with each other. Proposition 8.2.2. The endomorphism F ℓ (ˆ σ ℓ ) is invertible in End Q ( Q ⊗ Q Q ( µ m )) , where Q = Frac Λ E (Γ) .Proof. Clear, since the roots of the characteristic polynomial of ˆ σ ℓ on Q ( µ m ) are scalars. (cid:3) ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 73 For each m ∈ A , let us define an element x ′ m ∈ Q ⊗ Q Q ( µ m ) by x ′ m = Y ℓ | m F ℓ (ˆ σ ℓ ) − G ℓ (ˆ σ ℓ ) · (1 ⊗ ζ m ) . Proposition 8.2.3. If ℓ ∤ m , then we have tr mℓm ( x ′ mℓ ) = σ − ℓ F ℓ ( σ ℓ ) − (( ℓ − G ℓ ( σ ℓ ) − ℓF ℓ ( σ ℓ )) x ′ m . Proof. This is a straightforward generalization of (one case of) Proposition 2.5 of [Ots09]. We define H ℓ ( X ) = G ℓ ( X ) − F ℓ ( X ) X , so we have F ℓ (ˆ σ ℓ ) − G ℓ (ˆ σ ℓ ) = 1 + H ℓ (ˆ σ ℓ ) F ℓ (ˆ σ ℓ ) − ˆ σ ℓ in End Q Q ⊗ Q Q ( µ mℓ ). The operator ˆ σ v commutes with tr mℓm whenever v = ℓ . Hencetr mℓm ( x ′ mℓ ) = tr mℓm Y v | mℓ F v (ˆ σ v ) − G v ( σ v ) ζ mℓ = Y v | m F v (ˆ σ v ) − G v (ˆ σ v ) tr mℓm (cid:0) F ℓ (ˆ σ ℓ ) − G ℓ ( σ ℓ ) ζ mℓ (cid:1) = Y v | m F v (ˆ σ v ) − G v (ˆ σ v ) tr mℓm (cid:0)(cid:0) H ℓ (ˆ σ ℓ ) F ℓ (ˆ σ ℓ ) − ˆ σ ℓ (cid:1) ζ mℓ (cid:1) = Y v | m F v (ˆ σ v ) − G v (ˆ σ v ) (cid:2) tr mℓm ( ζ mℓ ) + tr mℓm (cid:0) H ℓ (ˆ σ ℓ ) F ℓ (ˆ σ ℓ ) − ζ m (cid:1)(cid:3) = Y v | m F v (ˆ σ v ) − G v (ˆ σ v ) (cid:2)(cid:0) − σ − ℓ ζ m (cid:1) + ( ℓ − (cid:0) H ℓ (ˆ σ ℓ ) F ℓ (ˆ σ ℓ ) − ζ m (cid:1)(cid:3) = (cid:0) − σ − ℓ + ( ℓ − H ℓ ( σ ℓ ) F ℓ ( σ ℓ ) − (cid:1) x ′ m (where we have dropped the hats, since ˆ σ ℓ acts on Q ⊗ Q ( µ m ) as the usual Frobenius σ ℓ ). Since σ ℓ H ℓ ( σ ℓ ) = G ℓ ( σ ℓ ) − F ℓ ( σ ℓ ), we have − σ − ℓ + ( ℓ − H ℓ ( σ ℓ ) F ℓ ( σ ℓ ) − = σ − ℓ F ℓ ( σ ℓ ) − ( − F ℓ ( σ ℓ ) + ( ℓ − σ ℓ H ℓ ( σ ℓ ))= σ − ℓ F ℓ ( σ ℓ ) − ( − F ℓ ( σ ℓ ) + ( ℓ − G ℓ ( σ ℓ ) − F ℓ ( σ ℓ )))= σ − ℓ F ℓ ( σ ℓ ) − (( ℓ − G ℓ ( σ ℓ ) − ℓF ℓ ( σ ℓ )) . which gives the formula stated above. (cid:3) Corollary 8.2.4. If we are given, for each m ∈ A , an element z m ∈ Q ⊗ Λ(Γ) H ( Q ( µ mp ∞ ) , V ) satisfying cores mℓm ( z mℓ ) = F ℓ ( σ − ℓ ) z m for each m and each prime ℓ ∤ m , ℓ / ∈ Σ , and we define x m = x ′ m v ∈ D mp ∞ for some fixed v ∈ D , thenwe have the relation pr mℓm t mℓ ( x mℓ , z mℓ ) = σ ℓ (cid:0) ( ℓ − G ℓ ( σ − ℓ ) − ℓF ℓ ( σ − ℓ ) (cid:1) t m ( x m , z m ) . By base extension we may regard t m ( x m , − ) as a map V m ) M m → M m , where M m = Q ⊗ Λ(Γ) H ( Q ( µ mp ∞ ) , V ) , so it makes sense to evaluate t m ( x m , − ) against a rank 2 Euler–Iwasawa system.We now specialize to the case where V = V L λ ( f, g ) ∗ , for some weight 2 eigenforms ( f, g ) of levelsdivisible only by primes in Σ − { p } . We take G ℓ ( X ) = 1 − ε ℓ ( f ) ε ℓ ( g ) X and F ℓ ( X ) = P ℓ ( ℓ − X ) asbefore. Choose a p -stabilization ( α, γ ) of f and g , and let v = v α ⊗ v γ be the obvious ϕ -eigenvector in D cris ( V ∗ ) of eigenvalue αγ . Proposition 8.2.5. Let ( w m ) m ≥ be an Euler–Iwasawa system of rank 2 for ( T, A, Σ) , for some lattice T in V , and let v m = t m ( x m , w m ) . Then we have v m ∈ H h (Γ) ⊗ Λ(Γ) Q ⊗ Λ(Γ) H ( Q ( µ mp ∞ ) , V ) , where h = v p ( αγ ) ; and the elements v m satisfy the compatibility relation cores mℓm v mℓ = σ ℓ (cid:0) ( ℓ − G ℓ ( σ − ℓ ) − ℓF ℓ ( σ − ℓ ) (cid:1) v m . Note that the growth condition H h (Γ) is consistent with what we have seen for the elements z f α ,g γ ,Npm (cf. Theorem 6.8.4) and the compatibility condition between levels m and mℓ is consistent with Theorem3.4.1. This suggests the following conjecture: Conjecture 8.2.6. Then there exists a rank 2 Euler–Iwasawa system ( w m ) for ( T O λ ( f, g ) ∗ , A, Σ) withthe property that for all m ∈ A , and all choices of p -stabilizations ( α, γ ) of ( f, g ) , the Iwasawa cohomologyclass z f α ,g γ ,Npm of Theorem 6.8.4 is given by z f α ,g γ ,Npm = t m ( x m , w m ) in the notation above. This gives a conceptual explanation for the (somewhat surprising) growth and compatibility propertiesof the generalized Beilinson–Flach elements in the context of Perrin-Riou’s theory of higher-rank Eulersystems. The authors would like to express their cautious hope that similar rank 1 “shadows” of higherrank Euler systems might also exist in other contexts. Remark . Note that it is implicit in this conjecture that the elements t m ( x m , w m ) have no poles(except possibly at the trivial character), so the singularities of t m , at the characters where one of the F ℓ (ˆ σ ℓ ) for ℓ | m fails to be invertible, must be “cancelled out” by zeroes of w m . Appendix A. Ancillary results A.1. Fixed points of double cosets. Here we shall prove a result that is used in the proof of Theorem3.4.1 above.Let Γ be a discrete subgroup of PSL ( R ). Recall that a fundamental domain for Γ is a closed subset D of H such that • D is equal to the closure of its interior D ◦ , • S γ ∈ Γ γD = H , • γD ◦ ∩ D ◦ = ∅ for all non-identity elements γ ∈ Γ.We assume henceforth that Γ is a Fuchsian group of the first kind , i.e. that Γ admits a fundamentaldomain D with finite hyperbolic area. We shall say that a fundamental domain D is polygonal if D isthe region bounded by a finite number of geodesic arcs in H ; it is known that every Γ admits a polygonalfundamental domain. Lemma A.1.1. Let D be a Dirichlet domain for Γ , and let E = αD where α lies in the commensurator Comm(Γ) . Then there are only finitely many γ ∈ Γ such that αD ∩ γD = ∅ .Proof. It is clear that αD is a Dirichlet domain for α Γ α − . In particular, it is polygonal. Hence it can bedecomposed as the union of a compact set M and a finite number N i of “cusp neighbourhoods”, whichare subsets bounded by two geodesics intersecting at a vertex at infinity, which is a parabolic point x i of α Γ α − on the boundary P ( R ), and an arc of a Euclidean circle tangent to the real line at x i .Since M is compact, it can intersect only finitely many Γ-translates of D (cf. [Kat92, Theorem 3.5.1]).Moreover, since α ∈ Comm( D ), the sets of parabolic points of Γ and α Γ α − are the same; so for eachvertex-at-infinity x of αD , we may choose some γ ∈ Γ which maps a vertex-at-infinity y i of D to x i ,and it is clear that N i is contained in a finite union of translates of γγ ′ D where γ ′ lies in the stabilizerof y i . Each of these, in turn, intersects finitely many other translates of D (since D has finitely manysides). (cid:3) ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 75 Lemma A.1.2. Let Γ be Fuchsian group of the first kind, and let X ⊂ Comm(Γ) be a finite union ofdouble cosets Γ α Γ . Then the set Fix( X ) = { u ∈ H : γu = u for some γ ∈ X , γ = 1 } is a finite union of Γ -orbits in H .Proof. Since X − { id } is preserved by conjugation by Γ, the set Fix( X ) is a union of orbits of Γ. So itsuffices to show that Fix( X ) ∩ D is finite, where D is a Dirichlet domain for Γ.We claim that there are only finitely many x ∈ X such that xD ∩ D = ∅ . From Lemma A.1.1, weknow that for each α ∈ G there are finitely many γ ∈ Γ such that γD ∩ αD = ∅ , and hence finitelymany x ∈ Γ α such that xD ∩ D = ∅ . Since X is the union of finitely many left cosets Γ α i , this impliesthe claim.However, each non-identity element in the finite set { x ∈ X : xD ∩ D = ∅ } can only have finitelymany fixed points in H , and in particular in D ; so Fix( X ) ∩ D is finite, as required. (cid:3) Lemma A.1.3. Let Γ , Γ be commensurable Fuchsian groups of the first kind. Then the set { u ∈ H : ∃ c ∈ Γ , d ∈ Γ such that cd = 1 and cdu = u } is a finite union of orbits under Γ ∩ Γ .Proof. This follows from the previous lemma applied to Γ = Γ ∩ Γ and X = Γ Γ . (cid:3) (Note that if Γ = Γ , or more generally if the group generated by Γ and Γ is Fuchsian, thisgeneralizes the well-known result that Fuchsian groups of the first kind have finitely many elliptic pointsin Γ.)In particular, we have the following: Proposition A.1.4. Let Γ , Γ be commensurable Fuchsian groups of the first kind. Then the naturalmap (Γ ∩ Γ ) \H → (Γ \H ) × (Γ \H ) is injective away from a finite subset of its domain.Proof. Let z, z ′ be two points of H such that z ′ ∈ Γ z and z ′ ∈ Γ z . Then we may write z ′ = γ z forsome γ ∈ Γ and z ′ = γ z for some γ ∈ Γ .Hence γ − γ z = z . So either z lies in the finite subset Fix(Γ Γ ) of (Γ ∩ Γ ) \H , or γ − γ = 1, inwhich case z and z ′ are clearly in the same orbit under Γ ∩ Γ . (cid:3) A.2. Unbounded Iwasawa cohomology. In this section, we shall consider inverse systems of cohomol-ogy classes in Z × p -extensions which are not bounded (as in the usual definition of Iwasawa cohomology)but satisfy a weaker growth condition.Let K be a finite extension of either Q or Q p . If K is global, suppose that either p = 2, or K has noreal places. Notation. In order to handle the two cases in a uniform manner, we shall adopt a notation that is slightlyabusive: for T a Z p -representation of Gal( ¯ K/K ), the notation H i ( K, T ) will mean either H i ( K, T ) asdefined above if G is local, or what we previously called H iS ( K, T ) if K is global, where S is some fixedfinite set of places containing all infinite places and all those dividing p . In the latter case, we will assumethat S contains all primes at which T is ramified.As before, we let K n = K ( µ p n ) and K ∞ = S n K n , and define Iwasawa cohomology groups H i Iw ( K ∞ , T )as the inverse limit of the H i ( K n , T ) with respect to corestriction, with their natural module structureover Λ = Λ Z p (Γ). Proposition A.2.1 (Nekovar) . For any j ∈ { , , } , we have a short exact sequence (22) 0 ✲ H j Iw ( K ∞ , T ) Γ n ✲ H j ( K n , T ) ✲ H j +1Iw ( K ∞ , T ) Γ n ✲ Proof. This is Corollary 8.4.8.2 of [Nek06]. We briefly recall the proof. There are natural isomorphisms H i Iw ( K ∞ , T ) ∼ = H i ( K, Λ ⊗ Z p T ) H i ( K n , T ) ∼ = H i ( K, Z p [Γ / Γ n ] ⊗ Z p T ) (for a suitable Λ-linear action of G K on the tensor products); see Proposition 8.3.5 of op.cit. . Then theresult above follows from the long exact cohomology sequence of K -cohomology attached to the shortexact sequence of Λ[ G K ]-modules0 ✲ Λ ⊗ Z p T [ γ n ] − ✲ Λ ⊗ Z p T ✲ Z p [Γ / Γ n ] ⊗ Z p T ✲ . (cid:3) Proposition A.2.2 (Perrin-Riou) . There is an exact sequence ✲ T G K ∞ ✲ H ( K ∞ , T ) ✲ Hom Λ ( H ( K ∞ , V /T ) ∨ , Λ) ι ✲ (finite) ✲ , where ι signifies that the Λ -module structure is composed with the automorphism γ γ − . In both cases,this exact sequence identifies T G K ∞ with the Λ -torsion submodule of H ( K ∞ , T ) .Proof. The local case is [PR92, Proposition 2.1.6]; note that in the local situation Tate duality furnishesan isomorphism H ( K ∞ , V /T ) ∨ ∼ = H ( K ∞ , T ∗ (1)) and the middle map can be interpreted as Perrin-Riou’s pairing H ( K ∞ , T ) × H ( K ∞ , T ∗ (1)) → Λ. The global case is [PR95, Lemma 1.3.3]. (cid:3) The main object of study in this section is the following module. Let V = Q p ⊗ Z p T . Definition A.2.3. For K, T, V as above, and ≤ r < , let Y r ( K ∞ , V ) be the space of sequences ( c n ) n ≥ ∈ lim ←− n H ( K n , V ) such that there exists δ < ∞ independent of n for which p ⌊ rn ⌋ + δ c n is in theimage of H ( K n , T ) in H ( K n , V ) . Proposition A.2.4. For all ≤ r < , the natural map λ r : H r (Γ) ⊗ Λ Q p (Γ) H ( K ∞ , V ) → lim ←− n H ( K n , V ) has image contained in Y r ( K ∞ , V ) .Proof. This is clear from the definition of H r (Γ). (cid:3) Remark A.2.5 . The map λ r is not necessarily injective, even for r = 0 (where H r (Γ) is just Λ ⊗ Q p ). Acounterexample is provided by the representation T = Z p (1). Then the cocycle c n given by σ χ ( σ ) − p n is well-defined as an element of H ( K n , T ) (for either local or global K ). The sequence ( c n ) defines anelement of H ( K ∞ , T ) which is not p -torsion, and thus is non-zero as an element of H ( K ∞ , V ). But p n c n is a coboundary for all n , so the image of c n in H ( K n , V ) is zero for all n . Thus ( c n ) lies in thekernel of the above map. Proposition A.2.6. The kernel of λ r is contained in H ( K ∞ , V ) tors ∼ = V G K ∞ .Remark A.2.7 . We note first that this statement does make sense, since for any Λ Q p (Γ)-torsion module M , tensoring with 1 ∈ H r (Γ) gives an isomorphism H r (Γ) ⊗ Λ Q p (Γ) M ∼ = M . Proof. Tensoring (22) with Q p , we find that the map H ( K ∞ , V ) Γ n → H ( K n , V )is injective. Thus the kernel of λ r consists of those elements lying in \ n ≥ ( γ n − (cid:16) H r (Γ) ⊗ Λ Q p (Γ) H ( K ∞ , V ) (cid:17) . Since H ( K ∞ , V ) is a finitely-generated module over the subring Λ Q p (Γ ) ⊂ Λ Q p (Γ), which is a PID,we may write it as the direct sum of its torsion submodule and a complementary free submodule. Since r < 1, we find that \ n ≥ ( γ n − H r (Γ) = 0 , and hence the kernel of λ r is contained in the torsion part of H ( K ∞ , V ), which is equal to V G K ∞ byProposition A.2.2. (cid:3) Remark A.2.8 . Although we shall not need this, it clearly follows that the kernel of λ r is equal to S n ≥ ( γ n − V G K ∞ , which is the unique Γ-invariant complement of S n H ( K n , V ) in H ( K ∞ , V ). Proposition A.2.9. Let K be a p -adic field and suppose that V G K ∞ = 0 . Then the map H r (Γ) ⊗ Λ Q p (Γ) H ( K ∞ , V ) → Y r ( K ∞ , V ) is an isomorphism. ULER SYSTEMS FOR RANKIN–SELBERG CONVOLUTIONS OF MODULAR FORMS 77 Proof. By Proposition A.2.2, our hypotheses imply that H ( K ∞ , V ) is a torsion-free Λ Q p (Γ)-module;hence it is free, since Λ Q p (Γ) is a finite product of principal ideal domains (and the ranks of the Γ tors -isotypical direct summands of H ( K ∞ , V ) are all equal). Thus there exists a free basis x , . . . , x d of H ( K ∞ , V ), where d = dim Q p ( V ).For each n , the cokernel of the projection map H ( K, T ) → H ( K n , T )is finite, and its order is bounded independently of n . (In fact, the cokernel of this map is isomorphic tothe Γ n -invariants of H ( K ∞ , T ) ∼ = H ( K ∞ , ( V /T ) ∗ (1)) ∨ , and H ( K ∞ , ( V /T ) ∗ (1)) is finite, since H ( K ∞ , V ∗ (1)) = H ( K ∞ , V ) ∗ (1) = 0.) Thus the map H ( K ∞ , V ) → H ( K n , V )is surjective for all n , and there is ν < ∞ independent of n such that the Z p -submodule spanned by theimages of x , . . . , x d in H ( K n , V ) contains p ν · H ( K n ,T )torsion .Consequently, given any sequence ( c n ) n ≥ ∈ Y r ( K ∞ , V ), we have for each n uniquely determinedelements b ( n )1 , . . . , b ( n ) d ∈ Q p [Γ / Γ n ] such that P di =1 b ( n ) i x ( n ) i = c n , where x ( n ) i is the image of x i in H ( K n , V ). Also, for each i the sequence ( b ( n ) i ) n ≥ is compatible under projection (by uniqueness), andits valuation is bounded below by −⌊ rn ⌋ − δ − ν ; so (as r < 1) there is a unique element b i ∈ H r (Γ)whose image at level n is b ( n ) i for all n . Then it is clear that c = P i b i ⊗ x i ∈ H r (Γ) ⊗ Λ Q p (Γ) H ( K ∞ , V )is a preimage of ( c n ) n ≥ ; by the previous proposition, it is unique. (cid:3) In the global case we cannot prove quite such a strong result, as we do not have such good controlover H ( K ∞ , T ); the following rather more specific result (which applies to both local and global cases)will suffice for our purposes: Proposition A.2.10. Let r < , and suppose T has the structure of a module over O E , for some finiteextension E/ Q p , and that T G K ∞ = 0 . Let α ∈ O E such that v p ( α ) ≤ r , and suppose we are givenelements x n ∈ H ( K n , T ) for n ≥ satisfying cores n +1 n ( x n +1 ) = αx n . Then there is a unique element x ∈ H r (Γ) ⊗ Λ H ( K ∞ , T ) whose image in H ( K n , V ) is equal to α − n c n for all n .Proof. We claim that the hypotheses of the theorem force each c n to be the image of an element of H ( K ∞ , V ) Γ n . To prove this, we shall argue much as in the proof of Proposition 6.6.2. We beginby noting that H ( K ∞ , T ) is a finitely-generated Λ-module, and the subgroups M n := H ( K ∞ , T ) Γ n are Λ-submodules (since Γ is abelian). As Λ is a Noetherian ring, the ascending chain of submodules( M n ) n ≥ must eventually stabilize; that is, there is an n such that M n = M n for all n ≥ n .The corestriction map H ( K n +1 , T ) → H ( K n , T ) corresponds to the trace map M n +1 → M n ; when n ≥ n this is simply multiplication by p on M n . Since v p ( α ) < 1, we deduce as in Proposition 6.6.2that for all n ≥ 0, the image of x n is contained in the torsion submodule of M n . Inverting p , the torsionis killed, and the image of x n in H ( K ∞ , V ) Γ n is 0; so x n lies in the submodule H ( K ∞ , V ) Γ n ⊆ H ( K n , V ).Now let us choose a basis of the free Λ Q p (Γ)-module H ( K ∞ , V ). In order to apply the argu-ment of the previous proposition, we need only check that the order of the torsion subgroup of M n is bounded independently of n ; but this is immediate from the fact that the M n stabilize for large n (and are all finitely-generated as Z p -modules). 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(9)(Lei) Department of Mathematics and Statistics, Burnside Hall, McGill University, Montreal, QC, CanadaH3A 2K6 E-mail address : [email protected] (Loeffler) Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK E-mail address : [email protected] (Zerbes) Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK E-mail address ::