Mazur-Tate elements of non-ordinary modular forms
aa r X i v : . [ m a t h . N T ] J un MAZUR–TATE ELEMENTS OF NON-ORDINARY MODULARFORMS
ROBERT POLLACK AND TOM WESTON
Abstract.
We establish formulae for the Iwasawa invariants of Mazur–Tateelements of cuspidal eigenforms, generalizing known results in weight 2. Ourfirst theorem deals with forms of “medium” weight, and our second deals withforms of small slope. We give examples illustrating the strange behavior whichcan occur in the high weight, high slope case. Introduction
Fix an odd prime p , and let f denote a cuspidal eigenform of weight k ≥ ( N ) with p ∤ N . Throughout this introduction, we assume forsimplicity that f has rational Fourier coefficients. Let ρ f : G Q → GL ( F p ) denotethe associated residual Galois representation which we assume to be irreducible. If f is a p -ordinary form, then the p -adic L -function L p ( f ) is an Iwasawa function,and one can associate to f (analytic) Iwasawa invariants µ ( f ) = µ ( L p ( f )) and λ ( f ) = λ ( L p ( f )).If f is p -non-ordinary, then the situation is quite different as L p ( f ) is no longer anIwasawa function, and one does not have associated µ - and λ -invariants. However,when f has weight 2, constructions of Kurihara and Perrin-Riou produce pairs of µ - and λ -invariants denoted by µ ± ( f ) and λ ± ( f ) (see also [15] when a p ( f ) = 0).These invariants are defined by working with the Mazur–Tate elements θ n ( f ) ∈ Z p [ G n ]attached to f ; here G n = Gal( Q n / Q ) where Q n is the n th layer of the cyclotomic Z p -extension of Q . These elements interpolate the algebraic part of special valuesof the L -series of f ; in fact, L p ( f ) can be reconstructed as a limit of the θ n ( f ).To define the Iwasawa invariants in the non-ordinary weight 2 case, one showsthat the sequences { µ ( θ n ( f )) } and { µ ( θ n +1 ( f )) } stabilize as n → ∞ ; the limitof these sequences are the invariants µ + ( f ) and µ − ( f ). For the λ -invariants, thesequence { λ ( θ n ( f )) } is unbounded, but grows in a regular manner: the invariants λ ± ( f ) have the property that (for sufficiently large n ) λ ( θ n ( f )) = q n + ( λ + ( f ) if 2 | nλ − ( f ) if 2 ∤ n, where q n = ( p n − − p n − + · · · + p − | np n − − p n − + · · · + p − p if 2 ∤ n. Mathematics Subject Classification.
Primary 11R23; Secondary 11F33.The first author was supported by NSF grant DMS-0701153 and a Sloan Research Fellowship.The second author was supported by NSF grant DMS-0700359.
In [8, 5, 9], the behavior of µ - and λ -invariants under congruences was studiedin the ordinary case for arbitrary weights and in the non-ordinary case in weight 2.For instance, in the ordinary case, it was shown that if the µ -invariant vanishes forone form, then it vanishes for all congruent forms. In particular, the vanishing of µ only depends upon the residual representation ρ = ρ f ; we write µ ( ρ ) = 0 if thisvanishing occurs. Completely analogous results hold in the weight 2 non-ordinarycase.The λ -invariant can change under congruences, but this change is expressible interms of explicit local factors. In fact, when µ ( ρ ) = 0, there exists some globalconstant λ ( ρ ) such that the λ -invariant of any form with residual representation ρ is given by λ ( ρ ) plus some non-negative local contributions at places dividing thelevel of the form.An analogous theory exists on the algebraic side for ordinary forms and weight 2non-ordinary forms. These invariants are built out of Selmer groups, and enjoy thecongruence properties described above. Furthermore, the Mazur–Tate elementsshould control the size and structure of the corresponding Selmer groups. Forinstance, in the non-ordinary case, the main conjecture predicts that(1) dim F p Sel p ( f / Q n )[ p ] = λ ( θ n ( f ))when µ ± ( f ) = 0 (see [9]). Here Sel p ( f / Q n ) is the p -adic Selmer group attached to f over the field Q n . Moreover, Kurihara [11, Conjecture 0.3] conjectures that theFitting ideals of these Selmer groups are generated by Mazur–Tate elements.Whether or not the equality in (1) extends to higher weight non-ordinary forms isunknown. Little is known about the size and structure of Selmer groups in this case.In this paper, we instead focus on the right hand side of (1), and via congruencesin the spirit of [8, 5], we attempt to describe the Iwasawa invariants of Mazur–Tateelements for non-ordinary forms which admit a congruence to some weight 2 form.1.1. Theorem for medium weight forms.
We offer the following theorem whichdescribes the Iwasawa invariants for “medium weight” modular forms (compare toCorollary 5.3 in the text of the paper). We note that the form g which appearsbelow is p -ordinary if and only if ρ f (cid:12)(cid:12) G Q p is reducible (see section 4.5). Theorem 1.
Let f be an eigenform in S k (Γ) which is p -non-ordinary, and suchthat (1) ρ f is irreducible of Serre weight 2, (2) 2 < k < p + 1 , (3) ρ f (cid:12)(cid:12) G Q p is not decomposable.Then there exists an eigenform g ∈ S (Γ) with a ℓ ( f ) ≡ a ℓ ( g ) (mod p ) for all primes ℓ = p such that if ρ f (cid:12)(cid:12) G Q p is reducible (resp. irreducible), then (1) µ ( θ n ( f )) = 0 for n ≫ ⇐⇒ µ ( g ) = 0 (resp. µ ± ( g ) = 0 ); (2) if the equivalent conditions of (1) hold, then λ ( θ n ( f )) = p n − p n − + λ ( g ) if ρ f (cid:12)(cid:12) G Q p is reducible ,q n − + λ - ε n ( g ) if ρ f (cid:12)(cid:12) G Q p is irreducible , for n ≫ ; here ε n equals the sign of ( − n . AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 3
Assuming that ρ g is globally irreducible, it is conjectured that µ ( g ) = 0 when g isordinary, and µ ± ( g ) = 0 when g is non-ordinary (see [7, 13]). Thus, the equivalentconditions of part (1) in Theorem 1 conjecturally hold. Further, by combiningTheorem 1 with the results of [5, 9], one can express λ ( θ n ( f )) in terms λ ( ρ ) andlocal terms at primes dividing N .We note that if either of the latter two hypotheses of Theorem 1 are removed,then there exist forms which do not satisfy the conclusions of this theorem. In fact,there are examples of modular forms with weight as small as p + 1 for which the µ -invariant of θ n ( f ) is positive for arbitrarily large n . In these examples, there isan obvious non-trivial lower bound on µ which we now explain.1.2. Lower bound for µ . We can associate to f its (plus) modular symbol ϕ f = ϕ + f ∈ H c (Γ , V k − ( Q p )) + ∼ = Hom Γ (cid:0) Div ( P ( Q )) , V k − ( Q p ) (cid:1) + ;here V k − ( Q p ) is the space of homogeneous polynomials of degree k − X , Y over Q p . We normalize this symbol so that it takes values in V k − ( Z p ),but not pV k − ( Z p ). We then define µ min ( f ) = µ +min ( f ) = min D ∈ ∆ ord p (cid:18) ϕ f ( D ) (cid:12)(cid:12)(cid:12) ( X,Y )=(0 , (cid:19) = min D ∈ ∆ ord p (cid:0) coefficient of Y k − in ϕ f ( D ) (cid:1) . Let G n = Gal( Q ( µ p n ) / Q ). The n th level Mazur–Tate element in Z p [ G n ] is givenby ϑ n ( f ) = X a ∈ ( Z /p n Z ) × c a · σ a with c a = coefficient of Y k − in ϕ f ( {∞} − { a/p n } )where σ a corresponds to a under the standard isomorphism G n ∼ = ( Z /p n Z ) × .The element θ n ( f ) is defined as the projection of ϑ n +1 ( f ) under the natural map Z p [ G n +1 ] → Z p [ G n ]. It follows immediately that µ min ( θ n ( f )) ≥ µ min ( f ) . Theorem for low slope forms.
The following theorem applies to modularforms of arbitrary weight, but with small slope (compare to Corollary 6.2). (Notethat this is a non-standard usage of the term slope as we are considering the valu-ation of the eigenvalue of T p as opposed to U p .) Theorem 2.
Let f be an eigenform in S k (Γ) such that (1) ρ f is irreducible of Serre weight 2, (2) 0 < ord p ( a p ) < p − , (3) ρ f (cid:12)(cid:12) G Q p is not decomposable.Then µ min ( f ) ≤ ord p ( a p ) . Further, there exists an eigenform g ∈ S (Γ) with a ℓ ( f ) ≡ a ℓ ( g ) (mod p ) for allprimes ℓ = p such that if ρ f (cid:12)(cid:12) G Q p is reducible (resp. irreducible), then (1) µ ( θ n ( f )) = µ min ( f ) for n ≫ ⇐⇒ µ ( g ) = 0 (resp. µ ± ( g ) = 0 ); ROBERT POLLACK AND TOM WESTON (2) if the equivalent conditions of (1) hold and n ≫ , then λ ( θ n ( f )) = p n − p n − + λ ( g ) if ρ f (cid:12)(cid:12) G Q p is reducible ,q n − + λ - ε n ( g ) if ρ f (cid:12)(cid:12) G Q p is irreducible . Note that the conclusions of Theorem 2 are the same as the conclusions of The-orem 1 except that the µ -invariants tend to µ min ( f ) rather than to 0.Condition (2) in Theorem 2 is necessary as there exist forms f of slope p − λ ( θ n ( f )) − q n +1 grow without bound for these forms. We will discuss these exceptional forms aftersketching the proofs of Theorems 1 and 2.1.4. Sketch of proof of Theorem 1.
First we consider the proof of Theorem 1in the case when k = p + 1. The map V p − ( Z p ) → F p P ( X, Y ) P (0 ,
1) (mod p )induces a map α : H c (Γ , V p − ( Z p )) → H c (Γ , F p )where Γ = Γ ( N p ). (Note that we have first composed with restriction to levelΓ .) The map α is equivariant for the full Hecke-algebra, where at p we let theHecke-algebra act on the source by T p and on the target by U p .By results of Ash and Stevens [1, Theorem 3.4a], we have that α ( ϕ f ) = 0. Thesystem of Hecke-eigenvalues of α ( ϕ f ) then arises as the reduction of the system ofHecke-eigenvalues of some eigenform h ∈ S (Γ ) (see [1, Proposition 2.5a]). Thisform h is necessarily non-ordinary, and since h has weight 2, it is necessarily p -old.Let g be the associated form of level Γ, and let ϕ g denote the reduction mod p of ϕ g , the (plus) modular symbol attached to g .A direct computation shows that ϕ g (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) is a Hecke-eigensymbol for the fullHecke-algebra with the same system of Hecke-eigenvalues as α ( ϕ f ). (The analogousstatement for ϕ g is false as this is an eigensymbol for T p and not for U p .) By mod p multiplicity one, we then have α ( ϕ f ) = ϕ g (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) . Note that we can insist upon an equality here as ϕ g is only well-defined up toscaling by a unit. This equality of modular symbols implies the following relationof Mazur–Tate elements: θ n ( f ) ≡ cor nn − ( θ n − ( g )) (mod p )where cor nn − : F p [ G n − ] → F p [ G n ] is the corestriction map. Since µ (cor nn − ( θ )) = µ ( θ ) and λ (cor nn − ( θ )) = p n − p n − + λ ( θ ) , Theorem 1 follows when k = p + 1.To illustrate how the proof proceeds for the remaining weights in the range p + 1 < k < p + 1, we consider the case when k = 2 p . If we can show that α ( ϕ f ) = 0, then the above proof goes through verbatim. So assume that ϕ f is inthe kernel of α . Then results of Ash and Stevens [1, Theorem 3.4c] imply that thereis an eigenform h ∈ S p − (Γ) such that ρ h ⊗ ω ∼ = ρ f ; here ω is the mod p cyclotomiccharacter. Since the weight of h is less than p + 1, Fontaine–Lafaille theory gives AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 5 an explicit description of ρ h (cid:12)(cid:12) G Q p . However, as long as ρ f (cid:12)(cid:12) G Q p is indecomposable,this description contradicts the fact that ρ f ∼ = ρ h ⊗ ω has Serre weight 2.We illustrate this argument once more when k = 3 p − α ( ϕ f ) = 0, then there exists an eigenform h ∈ S p − (Γ)such that ρ h ⊗ ω = ρ f . Since the weight of h is out of the Fontaine–Lafaille range,we cannot immediately determine the structure of ρ h (cid:12)(cid:12) G Q p . Instead, we considerthe modular symbol ϕ h ∈ H c (Γ , V p − ( Z p )). If α ( ϕ h ) = 0, then as before, h iscongruent to a weight 2 form, and we can describe the structure of ρ h (cid:12)(cid:12) G Q p . On theother hand, if α ( ϕ h ) = 0, then h is congruent to some eigenform h ′ ∈ S p − (Γ) suchthat ρ h ′ ⊗ ω ∼ = ρ h . As the weight of h ′ is small, we can determine the structure of ρ h ′ (cid:12)(cid:12) G Q p . In either case, this description implies that f cannot have Serre weight 2.For the remaining weights, one proceeds similarly, inductively decreasing theweight of the form being considered. These arguments work up until weight p + 1when in fact there can be an eigensymbol with Serre weight 2 in the kernel of α .1.5. Sketch of proof of Theorem 2.
For the proof of Theorem 2, we considerthe following Γ -stable filtration on V g ( Z p ):Fil r ( V g ) = g X j =0 b j X j Y g − j ∈ V g ( Z p ) : p r − j | b j for 0 ≤ j ≤ r − . One computes that if ϕ f takes values in j -th step of this filtration, then ord p ( a p ) isat least j . In particular, if r = ord p ( a p ) + 1, then ϕ f cannot take of all its values inFil r ( V k − ). The Jordan-Holder factors of V k − ( Z p ) / Fil r ( V k − ) are all isomorphicto F p with γ = (cid:0) a bc d (cid:1) ∈ Γ acting by multiplication by a i det( γ ) j for some i and j .The image of ϕ f in H c (Γ , V k − ( Z p ) / Fil r ( V k − )) is non-zero, and thus mustcontribute to the cohomology of one of these Jordan-Holder factors. In particular,there exists a congruent eigenform h of weight 2 on Γ ( N ) ∩ Γ ( p ). Again bycomputing the possibilities for ρ h (cid:12)(cid:12) G Q p , one can determine the exact Jordan-Holderfactor to which ϕ f contributes as long as r satisfies the bound in the hypothesis(2) of Theorem 2.As a result of this computation, one sees that the function∆ → F p D ϕ f ( D ) (cid:12)(cid:12)(cid:12) ( X,Y )=(0 , p µ min ( f ) (mod p )is a modular symbol of level Γ . One then proceeds as in the proof of Theorem 1 toconstruct a congruent weight 2 form, and then deduce the appropriate congruenceof Mazur–Tate elements.1.6. An odd example.
We close this introduction with one strange example. For p = 3, there is an eigenform f in S (Γ (17) , Q ) which is an eigenform of slope 5whose residual representation is isomorphic to the 3-torsion in X (17). (Note that X (17)[3] is locally irreducible at 3 as X (17) is supersingular at 3.) The form f does not satisfy the hypotheses of Theorems 1 or 2 as its weight and slope are toobig. For this form, we can show that µ min ( f ) = µ ( θ n ( f )) = 4 for all n ≥
0, and λ ( θ n ( f )) = p n − p n − + q n − for n ≥
2. This behavior of the λ -invariants is different from the patterns inTheorems 1 and 2 where the λ -invariants equal q n +1 = p n − p n − + q n − up toa bounded constant. As explained in section 7, this different behavior can beexplained in terms of the failure of multiplicity one at level N p r with r ≥ Outline.
The outline of the paper is as follows: we begin by reviewing thedefinition of Mazur–Tate elements and their relation to p -adic L -functions. As ourfocus will be on the Mazur–Tate elements, we then discuss finite-level Iwasawainvariants, recalling known results in the ordinary case (section 3) and the weight2 non-ordinary case (section 4). In section 5 (resp. section 6) we prove Theorem 1(resp. Theorem 2) on Iwasawa invariants of Mazur–Tate elements for non-ordinarymodular forms of medium weight (resp. low slope). In section 7, we explain indetail an example of this odd behavior of λ -invariants for a form of high weight andslope. Acknowledgements : We owe a debt to Matthew Emerton for numerous enlight-ening conversations on this topic. We heartily thank Kevin Buzzard for severalsuggestions which led to the proof of Theorem 2.
Notation:
Throughout the paper we fix an odd prime p . Let Z p denote the ringof integers of Q p , and for x ∈ Z p , let x denote the image of x in F p . For a finiteextension O of Z p , we write ̟ for a uniformizer of O , and F for its residue field.We fix an embedding Q ֒ → Q p . For an integer n , we write ε n for the sign of ( − n .2. Mazur–Tate elements of modular forms
Let f be a cuspidal eigenform of weight k on a congruence subgroup Γ = Γ ( N ).Our goal in this section is to define the p -adic Mazur–Tate elements ϑ n ( f ) attachedto f . These are elements of the group ring Z p [ G n ] for all n ≥
1; here G n = Gal( Q ( µ p n ) / Q ) ∼ = ( Z /p n Z ) × σ a ↔ a where σ a ( ζ ) = ζ a for ζ ∈ µ p n . The utility of these elements is that they allow one torecover normalized special values of twists of the L -function of f ; see Proposition 2.3for a precise statement.2.1. Mazur–Tate elements.
Let R be a commutative ring, and set V g ( R ) =Sym g ( R ) which we view as the space of homogeneous polynomials of degree g with coefficients in R in two variables X and Y . We endow V g ( R ) with a rightaction of GL ( R ) by( P | γ )( X, Y ) = P (( X, Y ) γ ∗ ) = P ( dX − cY, − bX + aY )for P ∈ V g ( R ) and γ ∈ GL ( R ).Let Γ ⊆ SL ( Z ) denote a congruence subgroup. Recall the canonical isomor-phism of Hecke-modules (see [1, Proposition 4.2]) H c (Γ , V g ( R )) ∼ = Hom Γ (cid:0) Div ( P ( Q )) , V g ( R ) (cid:1) where the target of the map equals the collection of additive maps (cid:8) ϕ : Div ( P ( Q )) → V g ( R ) : ϕ ( γD ) | γ = ϕ ( D ) for all γ ∈ Γ (cid:9) . AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 7
As this isomorphism is canonical, we will implicitly identify these two spaces fromnow on; we refer to them as spaces of modular symbols .For a modular symbol ϕ ∈ H c (Γ , V g ( R )), we define the associated Mazur–Tateelement of level n ≥ ϑ n ( ϕ ) = X a ∈ ( Z /p n Z ) × ϕ ( {∞} − { a/p n } ) (cid:12)(cid:12)(cid:12) ( X,Y )=(0 , · σ a ∈ R [ G n ] . When R contains Z p , we may decompose the Mazur–Tate elements ϑ n ( ϕ ) asfollows. Write G n +1 ∼ = G n × ( Z /p Z ) × with G n cyclic of order p n . Let ω : ( Z /p Z ) × → Z × p denote the usual embedding ofthe ( p − st roots of unity in Z p . For each i , 0 ≤ i ≤ p −
2, we obtain an inducedmap ω i : R [ G n +1 ] → R [ G n ], and we define θ n,i ( ϕ ) = ω i ( ϑ n +1 ( ϕ )). We simply write θ n ( ϕ ) for θ n, ( ϕ ).2.2. Modular forms.
One can associate to each eigenform f in S k (Γ , C ) a modularsymbol ξ f in H c (Γ , V k − ( C )) such that ξ f ( { r } − { s } ) = 2 πi Z rs f ( z )( zX + Y ) k − dz for all r, s ∈ P ( Q ); here we write { r } for the divisor associated to r ∈ Q . Thesymbol ξ f is a Hecke-eigensymbol with the same Hecke-eigenvalues as f .The matrix ι := (cid:0) − (cid:1) acts as an involution on these spaces of modular symbols,and thus ξ f can be uniquely written as ξ + f + ξ − f with ξ ± f in the ± ι . By a theorem of Shimura [19], there exists complex numbers Ω ± f such that ξ ± f takes values in V k − ( K f )Ω ± f where K f is the field of Fourier coefficients of f . Wecan thus view ϕ ± f := ξ ± f / Ω ± f as taking values in V k − ( Q p ) via our fixed embedding Q ֒ → Q p . Set ϕ f = ϕ + f + ϕ − f , which of course depends upon the choices of Ω + f andΩ − f .Throughout this paper it will be crucial that we have normalized these choicesof periods appropriately. For any ϕ ∈ H c (Γ , V k − ( Q p )), define || ϕ || := max D ∈ ∆ || ϕ ( D ) || where for P ∈ V k − ( Q p ), || P || is given by the maximum of the absolute values ofthe coefficients of P . Let O f denote the ring of integers of the completion of theimage of K f in Q p . Definition 2.1.
We say that Ω + f and Ω − f are cohomological periods for f (withrespect to our fixed embedding Q ֒ → Q p ), if || ϕ + f || = || ϕ − f || = 1; that is, if each of ϕ + f and ϕ − f takes values in V k − ( O f ), and each takes on at least one value with atleast one coefficient in O × f . Such periods clearly always exist for each f and arewell-defined up to scaling by elements α ∈ K f such that the image of α in Q p is a p -adic unit.We now write ϑ n ( f ) for the Mazur–Tate element ϑ n ( ϕ f ) computed with respectto cohomological periods. As before, we obtain Mazur–Tate elements θ n,i ( f ) ∈ O f [ G n ] ROBERT POLLACK AND TOM WESTON for each n ≥ i , 0 ≤ i ≤ p −
2. We simply write θ n ( f ) for θ n, ( f ). Remark 2.2.
We note that our choice of periods forces these Mazur–Tate elementsto have integral coefficients. This should be contrasted with the case of ellipticcurves where the choice of the N´eron period does not a priori guarantee integrality.The following proposition describes the interpolation property of Mazur–Tateelements for primitive characters.
Proposition 2.3. If χ is a primitive Dirichlet character of conductor p n > , then χ ( ϑ n ( f )) = τ ( χ ) · L ( f, χ, εf where ε f equals the sign of χ ( − .Proof. This proposition follows from [12, (8.6)]. (cid:3)
Remark 2.4.
We note that the classical Stickelberger element ϑ n = 1 p n X a ∈ ( Z /p n Z ) × a · σ − a ∈ Q [ G n ]has a similar interpolation property: for χ a primitive character on G n , χ ( ϑ n ) = − L ( χ, . Three-term relation.
Let π nn − : O [ G n ] → O [ G n − ] be the natural projec-tion, and let cor nn − : O [ G n − ] → O [ G n ] denote the corestriction map given bycor nn − ( σ ) = X τ στ ∈ G n τ for σ ∈ G n .We then have the following three-term relation among various Mazur–Tate ele-ments. Proposition 2.5. If p ∤ N , we have (3) π n +1 n ( θ n +1 ,i ( f )) = a p θ n,i ( f ) − p k − cor nn − ( θ n − ,i ( f )) . for n ≥ and any i .Proof. This proposition follows from [12, (4.2)] and a straightforward computation. (cid:3)
Some lemmas.
The following computations will be useful later in the paper.
Lemma 2.6.
For ϕ ∈ H c (Γ , V g ( O )) and n ≥ , we have θ n,i ( ϕ (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) ) = p g · cor nn − ( θ n − ,i ( ϕ )) . AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 9
Proof.
We have ϑ n ( ϕ (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) ) = X a ∈G n (cid:0) ϕ (cid:12)(cid:12) (cid:0) p
00 1 (cid:1)(cid:1) ( {∞} − { a/p n } ) (cid:12)(cid:12)(cid:12) ( X,Y )=(0 , · σ a = X a ∈G n ϕ (cid:0) {∞} − { a/p n − } (cid:1) (cid:12)(cid:12)(cid:12) ( X,Y )=(0 ,p ) · σ a = p g · X a ∈G n ϕ (cid:0) {∞} − { a/p n − } (cid:1) (cid:12)(cid:12)(cid:12) ( X,Y )=(0 , · σ a = p g · cor nn − ( ϑ n − ( ϕ )) . Projecting to O [ G n ] then gives the lemma. (cid:3) Lemma 2.7. If f is a newform on Γ ( N ) of weight k , then ϕ f (cid:12)(cid:12) (cid:0) − N (cid:1) = ± N k − ϕ f . Proof.
First note that as f is a newform, w N ( f ) = ± f , and thus N − k/ z − k f ( − /N z ) = ± f ( z ) . Computing, we have (cid:0) ξ f (cid:12)(cid:12) (cid:0) − N (cid:1)(cid:1) ( { r } − { s } )= ξ f ( {− /N r } − {− /N s } ) (cid:12)(cid:12) (cid:0) − N (cid:1) = 2 πi Z − /Nr − /Ns f ( z )( − N zY + X ) k − = 2 πiN Z rs f ( − /N z )( Y /z + X ) k − z − dz ( z
7→ − /N z )= ± N k/ − πi Z rs f ( z )( Y + zX ) k − dz = ± N k/ − ξ f ( { r } − { s } ) , and the lemma follows. (cid:3) The p -ordinary case In this section, we first introduce Iwasawa invariants in finite-level group alge-bras, and then analyze the µ - and λ -invariants of Mazur–Tate elements of p -ordinaryforms.3.1. Iwasawa invariants in finite-level group algebras.
Fix a finite integrallyclosed extension O of Z p and let Λ := lim ←− O [ G n ] denote the Iwasawa algebra. Given L ∈ Λ, we may define Iwasawa invariants of L as follows. Fix an isomorphismΛ ∼ = O [[ T ]] and write L = P ∞ j =0 a j T j ; we then define µ ( L ) = min j ord p ( a j ) λ ( L ) = min { j : ord p ( a j ) = µ ( L ) } . (This definition is independent of the choice of isomorphism Λ ∼ = O [[ T ]].) Here wenormalize ord p so that ord p ( p ) = 1. Note that under this normalization, if O is aramified extension of Z p , then µ ( L ) need not be in Z . In fact, Iwasawa invariants can also be defined in the finite-level group algebras O [ G n ]. Indeed, for θ ∈ O [ G n ], if write θ = P σ ∈ G n c σ σ , we then define µ ( θ ) = min σ ∈ G n ord p ( c σ ) . To define λ -invariants, let ̟ be a uniformizer of O , and set θ ′ = ̟ − a θ with a chosen so that µ ( θ ′ ) = 0. Let F be the residue field of O , and let θ ′ denote the(non-zero) image of θ ′ under the natural map O [ G n ] → F [ G n ]. All ideals of F [ G n ]are of the form I jn with I n the augmentation ideal; we then define λ ( θ ) = ord I n θ ′ = max { j : θ ′ ∈ I jn } . The following lemmas summarize some basic properties of these µ and λ -invariants.For more details, see [14, Section 4]. Lemma 3.1.
Fix L ∈ Λ and let L n denote the image of L in O [ G n ] . Then for n ≫ , we have µ ( L ) = µ ( L n ) and λ ( L ) = λ ( L n ) . Lemma 3.2.
For θ ∈ O [ G n − ] , we have (1) µ (cor nn − ( θ )) = µ ( θ ) , (2) λ (cor nn − ( θ )) = p n − p n − + λ ( θ ) . Lemma 3.3.
Fix θ ∈ O [ G n ] . (1) If µ ( π nn − ( θ )) = 0 , then µ ( θ ) = 0 . (2) If µ ( θ ) = 0 , then λ ( π nn − ( θ )) = λ ( θ ) . p -adic L -functions for p -ordinary forms. The Mazur–Tate elements θ n,i ( f )can be used to construct the p -adic L -function of f in the p -ordinary case. As thisconstruction motivates much of what we do here, we briefly digress to describe it.We first fix some notation for the remainder of this paper. Fix an integer N relatively prime to p and set Γ = Γ ( N ). Also set Γ = Γ ( N p ) and Γ = Γ ( N ) ∩ Γ ( p ).Let f be a weight k eigenform on Γ which is ordinary at p . Let α denote theunique unit root of x − a p x + p k − , and let f α denote the p -ordinary stabilizationof f to Γ . The three-term relation of Proposition 2.5 only has two terms when p divides the level, and so the Mazur–Tate elements θ n,i ( f α ) attached to f α satisfy π nn − ( θ n,i ( f α )) = α · θ n − ,i ( f α ) . If we set ψ n,i ( f α ) = 1 α n θ n,i ( f α ) , then { ψ n,i ( f α ) } is a norm-coherent sequence, and thus an element of Λ. Thiselement is exactly L p ( f, ω i ), the p -adic L -function of f , twisted by ω i , and computedwith respect to the periods Ω ± f α .3.3. Iwasawa invariants in the p -ordinary case. Let f continue to be a p -ordinary eigenform on Γ. Set µ ( f, ω i ) = µ ( L p ( f, ω i )) and λ ( f, ω i ) = λ ( L p ( f, ω i )).From the results of the last section and from Lemma 3.1, we have that(4) µ ( θ n,i ( f α )) = µ ( f, ω i ) and λ ( θ n,i ( f α )) = λ ( f, ω i )for n ≫
0. Thus, the Iwasawa invariants of these “ p -stabilized” Mazur–Tate ele-ments are extremely well-behaved in the ordinary case. AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 11
One would hope to deduce similar information about the Iwasawa invariants ofthe θ n,i ( f ). Unfortunately, they are not always as well-behaved as their p -stabilizedcounterparts as the following example illustrates. Example 3.4.
Let f = P n a n q n denote the newform of weight 2 on Γ (11) cor-responding to the elliptic curve E = X (11). If ϕ f denotes the reduction of themodular symbol attached to f modulo 5, we have(5) ϕ f = ϕ f (cid:12)(cid:12) ( ) . One verifies this relation by noting that ϕ f is (up to a non-zero scalar) the reductionof the Eisenstein boundary symbol defined by ϕ eis ( { r/s } ) = ( s,
11) = 11 otherwisewhere gcd( r, s ) = 1. Since ϕ eis satisfies (5) so does ϕ f .From repeated applications of Lemma 2.6 we now obtain θ n ( f ) ≡ cor nn − ( θ n − ( f )) ≡ · · · ≡ cor n ( θ ( f )) (mod 5) . Moreover, a direct computation shows that θ ( f ) is a unit and thus µ ( θ n ( f )) = 0 and λ ( θ n ( f )) = p n − n ≥
0. Hence, the λ -invariants in this case are unbounded. Here thededuction about λ -invariants comes from Lemma 3.2. We note that this is themaximal possible λ -invariant for any non-zero element of O [ G n ].There are several additional oddities in this example. First, the Iwasawa invari-ants of the θ n,i ( f α ) must behave nicely, and in fact, assuming the main conjecture,we have µ ( θ n ( f α )) = 1 and λ ( θ n ( f α )) = 0for all n ≥
0. In the process of p -stabilizing f , one considers the difference ϕ f − α ϕ f (cid:12)(cid:12) ( ). However, since a = 1, we have that α ≡ ± f and Ω ± f α differ by a multiple of 5, and we can choose them so thatΩ ± f α = 5Ω ± f . From this, one might expect the µ -invariants of the θ n ( f α ) to be lower than the µ -invariants of the θ n ( f ). However, numerically (for small n ) one sees that θ n ( f ) ≡ α cor nn − ( θ n − ( f )) (mod 5 )so that θ n ( f α ) = 15 (cid:18) θ n ( f ) − α cor nn − ( θ n − ( f )) (cid:19) is divisible by 5.Lastly, we mention that the N´eron period of the elliptic curve E in this caseagrees with Ω + f α up to a 5-unit.The oddities of the above example arise as the residual representation ρ f : G Q → GL ( F p )is globally reducible and µ ( L p ( f )) is positive. However, when we are not in thiscase, we verify now that the Iwasawa invariants of the θ n,i ( f ) are well-behaved. We first check that cohomological periods are unchanged under p -stabilizationso long as ρ f is irreducible. Recall that for a multiple M of N and a divisor r of M/N there is a natural degeneracy map B M/N,r : H c (Γ ( N ) , F p ) → H c (Γ ( M ) , F p ) ϕ ϕ (cid:12)(cid:12) ( r
00 1 ) . In particular, we define a map B p : H c (Γ , F p ) → H c (Γ , F p )by B p ( ψ , ψ ) B p, ( ψ ) + B p,p ( ψ ) . Theorem 3.5 (Ihara’s lemma) . The kernel of B p : H c (Γ , F p ) → H c (Γ , F p ) is Eisenstein.Proof. See [17]. (cid:3)
Lemma 3.6.
Let f be a p -ordinary newform of weight k and level Γ . Let α denotethe unit root of x − a p x + p k − , and let f α denote the p -stabilization of f to level Γ . If ρ f is globally irreducible and Ω ± f are a pair of cohomological periods for f ,then Ω ± f are also a pair of cohomological periods for f α .Proof. Since f α ( z ) = f ( z ) − βf ( pz ), where β is the non-unit root of x − a p x + p k − ,a direct computation shows that ξ ± f α = ξ ± f − α ξ ± f (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) and thus ξ + f α Ω + f + ξ − f α Ω − f = ϕ f − α ϕ f (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) . To establish the lemma we need to show that the reduction of the above symbol isnon-zero.For k >
2, suppose instead that ϕ f = α · ϕ f (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) . The only non-zero coefficientsin the values of ϕ f (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) occur in the X k − coefficients, and thus the same is truefor ϕ f . But, by Lemma 2.7, the vanishing of the coefficients of Y k − implies thevanishing of the coefficients of X k − . Thus, ϕ f = 0 which is a contradiction.For k = 2, consider the obviously injective map j : H c (Γ , F p ) → H c (Γ , F p ) ϕ (cid:18) ϕ, − α · ϕ (cid:19) . As ρ f is irreducible, by Ihara’s lemma, ( B p ◦ j )( ϕ f ) = 0, and thus, ϕ f − α · ϕ f (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) = 0as desired. (cid:3) We are now in a position to understand the Iwasawa invariants of θ n,i ( f ) for fp -ordinary with ρ f irreducible. AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 13
Proposition 3.7.
Assume that µ ( L p ( f, ω i )) = 0 and that ρ f is irreducible. Thenfor n ≫ , we have µ ( θ n,i ( f )) = 0 and λ ( θ n,i ( f )) = λ ( L p ( f, ω i )) . Proof.
By Lemma 3.6, ϕ f α = ϕ f − α ϕ f (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) , and hence,(6) θ n,i ( f α ) = θ n,i ( f ) − α cor nn − ( θ n − ,i ( f )) . Since we are assuming that µ ( θ n,i ( f α )) = 0 for n ≫
0, by (6) there exist sufficientlylarge n for which µ ( θ n,i ( f )) = 0.If k >
2, the argument proceeds as follows: the three-term relation of Proposi-tion 2.5 implies that if µ ( θ n,i ( f )) = 0 for one n , then µ ( θ m,i ( f )) = 0 for all m > n as desired. For the λ -invariants, by Lemma 3.2, λ (cor nn − ( θ n − ,i ( f ))) ≥ p n − p n − , and thus for n large enough, λ ( θ n,i ( f α )) < λ (cor nn − ( θ n − ,i ( f ))) . By (6) and (4), we then have λ ( θ n,i ( f )) = λ ( θ n,i ( f α )) = λ ( L p ( f, ω i ))as desired.For the case k = 2, one must argue more carefully because the three-term relationdoes not guarantee the vanishing of µ ( θ n,i ( f )) for all n if one knows the vanishingfor a single n . From (6) we do know that there exists sufficiently large n such that µ ( θ n,i ( f )) = 0. For such a sufficiently large n , (6) implies that λ ( θ n,i ( f )) = λ ( θ n,i ( f α ))as before. Hence, λ ( θ n,i ( f )) = λ (cor nn − ( θ n − ( f )))which implies that the reduction of these two elements are not equal. In partic-ular, by (3) of Proposition 2.5, µ ( π n +1 n ( θ n +1 ( f ))) = 0, and thus by Lemma 3.3, µ ( θ n +1 ( f )) = 0 as desired. Thus inductively, µ ( θ n,i ( f )) vanishes for all sufficientlylarge n . Finally, the statement about λ -invariants follows just as in the k > (cid:3) The non-ordinary case
In the non-ordinary case, the polynomial x − a p x + p k − has no unit root. Thus,the construction of p -adic L -functions described in section 3.2 does not yield integralpower series. Indeed, if α is either root of this quadratic, then dividing by powersof α introduces p -adically unbounded denominators. In the non-ordinary case, wetherefore focus our attention on the elements θ n,i ( f ), rather than on passing to alimit to construct an unbounded p -adic L -function. Known results in weight 2.
For modular forms of weight 2, the Iwasawainvariants of the associated Mazur–Tate elements were studied in detail by Kurihara[11] and Perrin-Riou [13]. We summarize their results in the following theorem.
Theorem 4.1.
Let i be an integer with ≤ i ≤ p − . (1) There exist constants µ ± ( f, ω i ) ∈ Z ≥ such that for n ≫ , µ ( θ n,i ( f )) = µ + ( f, ω i ) and µ ( θ n +1 ,i ( f )) = µ − ( f, ω i ) . (2) If µ + ( f, ω i ) = µ − ( f, ω i ) , then there exist constants λ ± ( f, ω i ) ∈ Z ≥ suchthat for n ≫ , λ ( θ n,i ( f )) = q n + ( λ + ( f, ω i ) i even λ − ( f, ω i ) i oddwhere q n = ( p n − − p n − + · · · + p − if n even p n − − p n − + · · · + p − p if n odd . Remark 4.2. (1) Perrin-Riou conjectured [13, 6.1.1] that µ + ( f, ω i ) = µ − ( f, ω i ) = 0 (see also[15, Conjecture 6.3]). This is an analogue of Greenberg’s conjecture on thevanishing of µ in the ordinary case. Indeed, Greenberg conjectures that µ vanishes whenever ρ f is irreducible; if f has weight 2 and is p -non-ordinary,then ρ f is always irreducible.(2) In [9], the assumption that µ + ( f, ω i ) = µ − ( f, ω i ) is removed, but the re-sulting formula for λ is slightly different in some cases when µ + ( f, ω i ) = µ − ( f, ω i ). However, since this case is conjectured to never occur, and inthis paper we will only use these formulas when µ ± ( f, ω i ) = 0, we will notgo further into this complication.(3) Unlike the ordinary case, the λ -invariants of these non-ordinary Mazur–Tate elements always grow without bound because of the presence of the q n term which is O( p n − ). Proof of Theorem 4.1.
In [13], it is proven that any sequence of elements θ n ∈O [ G n ] which satisfy the three-term relation of Proposition 2.5 satisfy the conclu-sions of this theorem. To give the spirit of these arguments, we give a proof herein the case when µ ( θ n,i ( f )) = 0 for n ≫ a p is not a unit, the three-term relation implies that(7) π n +1 n ( θ n +1 ,i ( f )) ≡ cor nn − ( θ n − ,i ( f )) (mod ̟ ) . Thus for n large enough we have λ ( θ n +1 ,i ( f )) = λ ( π n +1 n ( θ n +1 ,i ( f ))) (by Lemma 3.3)= λ (cor nn − ( θ n − ,i ( f ))) (by (7))= p n − p n − + λ ( θ n − ,i ( f )) (by Lemma 3.2) . Proceeding inductively then yields the theorem. (cid:3)
AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 15
Differences in weights greater than 2.
To compare with the case of weight2, we note that when f is of any weight k >
2, then the three-term relation takesthe form π n +1 n ( θ n +1 ,i ( f )) = a p θ n,i ( f ) − p k − cor nn − ( θ n − ,i ( f )) . The factor of p k − in the third term prevents the arguments of the previous sectionfrom going through. Indeed, the right hand side of the above equation vanishesmod ̟ , and one cannot make general deductions about the Iwasawa invariants ofsuch sequences unlike the case when k = 2. Instead, the strategy of this paper is tomake a systematic study of congruences between Mazur–Tate elements in weight k and in weight 2, and then to make deductions about Iwasawa invariants by invokingTheorem 4.1.4.3. Lower bound for µ . We note that there is an obvious lower bound for µ -invariants of Mazur–Tate elements in weights greater than 2. For ϕ ∈ H c (Γ , V k − ( O )),set µ min ( ϕ ) = min D ∈ ∆ ord p (cid:18) ϕ ( D ) (cid:12)(cid:12)(cid:12) ( X,Y )=(0 , (cid:19) ;thus µ min ( ϕ ) is the minimum valuation of the coefficients of Y k − in the values of ϕ . We write µ ± min ( f ) for µ min ( ϕ ± f ). Proposition 4.3.
We have that (1) µ ± min ( f ) < ∞ , (2) µ ( θ n,i ( f )) ≥ µ ε i min ( f ) .Proof. For the first part, if µ ± min ( f ) = ∞ , then θ n ( f ) vanishes for every n . ByProposition 2.3, we then have that L ( f, χ,
1) = 0 for every character χ of conductora power of p . But this contradicts a non-vanishing theorem of Rohrlich [18].The second part is immediate as θ n,i ( f ) is constructed out of the coefficients of Y k − of certain values of ϕ ε i f . (cid:3) Recall that ϕ f is normalized so that all of its values have coefficients which areintegral and at least one which is a unit. Thus, when k = 2, by definition µ ± min ( f ) isalways 0. However, when k >
2, it is possible that the coefficient of Y k − in everyvalue of ϕ f is a non-unit, and that the required unit coefficient occurs in anothermonomial; in this case µ ± min ( f ) would be positive.4.4. A map from weight k to weight . In this section we discuss a map fromweight k modular symbols to weight 2 modular symbols over F p introduced by Ashand Stevens in [1]. Set S ( p ) := (cid:8)(cid:0) a bc d (cid:1) ∈ M ( Z ) : ad − bc = 0 , p | c, p ∤ a (cid:9) ,g = k −
2, and V g = V g ( F ). Lemma 4.4.
For g > and g ≡ p − , the map V g −→ F P ( X, Y ) P (0 , is S ( p ) -equivariant, and thus induces a Hecke-equivariant map α : H c (Γ , V g ) −→ H c (Γ , F ) . Remark 4.5.
By Hecke-equivariant we mean the standard concept away from p ,and at p , we mean that α intertwines the action of T p on the source with U p on thetarget. Proof.
This lemma follows from a straightforward computation. We note that theHecke-equivariance at p follows from the fact that for P ∈ V g and g > (cid:0) P (cid:12)(cid:12) (cid:0) p
00 1 (cid:1)(cid:1) (cid:12)(cid:12)(cid:12) ( X,Y )=(0 , = 0 . (cid:3) The following simple lemma is the key to our approach of comparing Mazur–Tateelements of weight k and weight 2. Lemma 4.6.
For ϕ ∈ H c (Γ , V k − ( O )) , ϑ n ( α ( ϕ )) = ϑ n ( ϕ ) = ϑ n ( ϕ ) in F [ G n ] , where ϕ is the reduction of ϕ modulo ̟ .Proof. The first equality is true as these Mazur–Tate elements depend only on thecoefficients of Y k − in the values of ϕ , and the map α preserves these coefficients.The second equality is clear. (cid:3) The following lemma gives the analogue for modular symbols of the θ -operatorfor mod p modular forms. In what follows, if M is a S ( p )-module, then M (1) isthe determinant twist of M ; for a Hecke-module M , the Hecke-operator T n acts on M (1) by nT n . Lemma 4.7.
The map V g − p − (1) −→ V g P ( X, Y ) ( X p Y − XY p ) · P ( X, Y ) is S ( p ) -equivariant, and thus induces a Hecke-equivariant map θ : H c (Γ , V g − p − )(1) −→ H c (Γ , V g ) . Proof.
This is a straightforward computation. (cid:3)
Lastly, we note that the kernel of α is given by precisely the symbols with positive µ min . Lemma 4.8.
We have µ min ( ϕ ) > ⇐⇒ α ( ϕ ) = 0 .Proof. We have α ( ϕ ) = 0 if and only if all of the coefficients of Y k − occurring invalues of ϕ are divisible by ̟ , which is equivalent to µ min ( ϕ ) > (cid:3) Review of mod p representations of G Q p . For use in the following sections,we recall the possibilities for the local residual representation of a modular form ofsmall weight.Let ρ p : G Q p → GL ( F p ) be an arbitrary continuous residual representation ofthe absolute Galois group of Q p . If ρ p is irreducible, then ρ p (cid:12)(cid:12) I p is tamely ramified;here I p denotes the inertia subgroup of G Q p . Moreover, we have ρ p (cid:12)(cid:12) I p ∼ = ω t ⊕ ω pt where ω is a fundamental character of level 2 and 1 ≤ t ≤ p − p + 1 ∤ t .The integer t uniquely determines ρ p (cid:12)(cid:12) I p and we write I ( t ) for this representation.We note that I ( t ) ∼ = I ( pt ).If ρ p is reducible, then ρ p (cid:12)(cid:12) I p ∼ = (cid:18) ω a ∗ ω b (cid:19) where ω is the mod p cyclotomic character. Theorem 4.9.
Let f be an eigenform on Γ of weight k with ρ f irreducible. (1) If f is p -ordinary, then ρ f (cid:12)(cid:12) G Q p is reducible and ρ f (cid:12)(cid:12) I p ∼ = (cid:18) ω k − ∗ (cid:19) . (2) If f is p -non-ordinary and ≤ k ≤ p + 1 , then ρ f (cid:12)(cid:12) G Q p is irreducible and ρ f (cid:12)(cid:12) I p ∼ = I ( k − .Proof. See [4, Remark 1.3] for a thorough discussion of references for these results. (cid:3)
The following lemma will be useful later in the paper.
Lemma 4.10. If f is an eigenform in S (Γ , ω j , Q p ) with ρ f irreducible and ≤ j ≤ p − , then ρ f (cid:12)(cid:12) I p ∼ = I ( j + 1) if ρ f (cid:12)(cid:12) G Q p is irreducible, ω j +1 ∗ ! or ω ∗ ω j ! if ρ f (cid:12)(cid:12) G Q p is reducible.Proof. Consider the modular symbol ϕ f ∈ H c (Γ , F ) ( ω j ) . By [1, Theorem 3.4(a)] ,the system of eigenvalues of ϕ f occurs either in H c (Γ , V j ) or H c (Γ , V p − − j )( j ).By [1, Proposition 2.5], there then exists either an eigenform g ∈ S j +2 (Γ , Q p ) with ρ f ∼ = ρ g or an eigenform g ∈ S p +1 − j (Γ , Q p ) with ρ f ∼ = ρ g ⊗ ω j .In the first case, by Theorem 4.9, ρ f (cid:12)(cid:12) I p is equal to either I ( j + 1) or (cid:0) ω j +1 ∗ (cid:1) ,and, in the second case, ρ g (cid:12)(cid:12) I p is equal to either I ( p − j ) or (cid:0) ω p − j ∗ (cid:1) . In the lattercase, ρ f (cid:12)(cid:12) I p ∼ = I ( p − j ) ⊗ ω j ∼ = I ( p − j + j ( p + 1)) ∼ = I ( pj + p ) ∼ = I ( j + 1)or ρ f (cid:12)(cid:12) I p ∼ = (cid:0) ω p − j ∗ (cid:1) ⊗ ω j = ( ω ∗ ω j ) . (cid:3) The non-ordinary case for medium weights
In this section, we will prove a theorem about the Iwasawa invariants of Mazur–Tate elements in weights k such that 2 < k < p +1. For f ∈ S k (Γ , Q p ) a normalizedeigenform, recall that O := O f denotes the ring of integers of the finite extensionof Q p generated by the Fourier coefficients of f , and F := F f denotes the residuefield of O . We note that in [1] the cohomology groups considered are not taken with compact support.However, the difference between H and H c is Eisenstein, and since we are assuming our formshave globally irreducible Galois representations, this difference does not affect our arguments. Statement of theorem.Theorem 5.1.
Let f be an eigenform in S k (Γ , Q p ) which is p -non-ordinary, andsuch that (1) ρ f is irreducible, (2) 2 < k < p + 1 , (3) k ( ρ f ) = 2 and ρ f (cid:12)(cid:12) G Q p is not decomposable.Then (1) µ +min ( f ) = µ − min ( f ) = 0 , and (2) there exists an eigenform g ∈ S (Γ) with a ℓ ( f ) = a ℓ ( g ) for all primes ℓ = p ,and a choice of cohomological periods Ω f , Ω g such that ϑ n ( f ) = cor nn − (cid:0) ϑ n − ( g ) (cid:1) in F [ G n ] . Remark 5.2. (1) In the notation of the above theorem, we have that g is ordinary at p ifand only if ρ f (cid:12)(cid:12) G Q p is reducible. Indeed, ρ f ∼ = ρ g , and since g has weight 2,Theorem 4.9 implies that g is ordinary at p if and only if ρ g (cid:12)(cid:12) G Q p is reducible.(2) Hypothesis (3) is equivalent to assuming that ρ f (cid:12)(cid:12) I p is isomorphic to either I (1) or ( ω ∗ ) with ∗ neither 0 nor tr`es-ramifi´ee.(3) Theorem 5.1 can fail for weights as low as p + 1. For example, there is anewform f ∈ S (Γ (17)) which is non-ordinary at p = 3 and congruent tothe unique normalized newform g ∈ S (Γ (17)). The form g is non-ordinaryat 3, and thus ρ f (cid:12)(cid:12) G Q p ∼ = ρ g (cid:12)(cid:12) G Q p is irreducible by Theorem 4.9. In particular,hypotheses (1) and (3) are satisfied; however, for this form, one computesthat µ +min ( f ) = 1. (We note that determining µ ± min ( f ) for a particular form f is a finite computation.)Possibly such counter-examples are common for following reason: let g ∈ S (Γ , Q p ) denote any eigenform which satisfies hypotheses (1) and(3). Consider θ p − ( ϕ g ) which is an eigensymbol in H c (Γ , V p − ). By [1,Proposition 2.5], there exists an eigenform f ∈ S p +1 (Γ , Q p ) whose systemof Hecke-eigenvalues reduces to those of θ p − ( ϕ g ). Thus, by Fermat’s littletheorem, a ℓ ( f ) = ℓ p − a ℓ ( g ) = a ℓ ( g )for all ℓ = p . Moreover, a p ( f ) = a p ( g ) since both are 0. Thus, ϕ f and θ p − ( ϕ g ) have the same system of Hecke-eigenvalues for the full Hecke-algebra. A strong enough mod p multiplicity one theorem (which is notcurrently known, and may not be always be true) would then imply equalityof these two symbols up to a constant. Thus, ϕ f is in the image of θ , andby Lemma 4.8, we would then have that µ ± min ( f ) > ρ f (cid:12)(cid:12) G Q p is not decomposable is necessary. For example,there is a newform f ∈ S (Γ (21)) which is non-ordinary at 5 and congru-ent to the unique normalized newform g ∈ S (Γ (21)). In this example, ρ f is irreducible, ρ f (cid:12)(cid:12) G Q p is decomposable, and µ ± min ( f ) > AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 19
Possibly such counter-examples are again common for a similar reasonas in the previous remark. Take g ∈ S (Γ , Q p ) with ρ g irreducible and ρ g (cid:12)(cid:12) G Q p decomposable. By Gross’ tameness criterion [10], there exists a form h ∈ S p − (Γ , Q p ) such that ρ h ⊗ ω ∼ = ρ g . The associated eigensymbol ϕ h is in H c (Γ , V p − ( Z p )), and thus θ ( ϕ h ) is in H c (Γ , V p − ( Z p )). By [1, Proposition2.5], there exists f ∈ S p (Γ , Q p ) whose system of Hecke-eigenvalues reducesto those of θ ( ϕ h ). In particular, ρ f ∼ = ρ h ⊗ ω ∼ = ρ g , and thus f satisfieshypotheses (1), k ( ρ f ) = 2, and ρ f (cid:12)(cid:12) G Q p decomposable.Note that θ ( ϕ h ) and ϕ f have the same system of Hecke-eigenvalues.Thus, as before, a strong enough mod p multiplicity one result would giveequality of these symbols up to a constant. In particular, we would obtainthat ϕ f is in the image of θ , and by Lemma 4.8, µ ± min ( f ) > ρ f (cid:12)(cid:12) G Q p remains a difficult one.Partial results exist when the weight k is not too large. For instance, if k = p + 1 and f is non-ordinary, then by a result of Edixhoven [6], ρ f (cid:12)(cid:12) G Q p isautomatically irreducible and isomorphic to I (1). More recently, Berger [3]showed that if k = 2 p , then ρ f (cid:12)(cid:12) G Q p is irreducible if and only if ord p ( a p ) = 1.Moreover, ρ f (cid:12)(cid:12) I p = I (1) if 0 < ord p ( a p ) < I (2 p −
1) if ord p ( a p ) > ω ∗ ) or ( ∗ ω ) if ord p ( a p ) = 1Unfortunately, even in this small weight, we do not know how to determinewhich representation occurs in the last of these three cases solely from thevalue of ord p ( a p ), and, in particular, we cannot determine the value of k ( ρ f ).In the following corollary we maintain the hypotheses and notation of Theorem5.1. Corollary 5.3. If ρ f (cid:12)(cid:12) G Q p is reducible (resp. irreducible), then (1) µ ( θ n,i ( f )) = 0 for n ≫ ⇐⇒ µ ( g, ω i ) = 0 (resp. µ ± ( g, ω i ) = 0 ); (2) if the equivalent conditions of (1) hold and n ≫ , then λ ( θ n,i ( f )) = p n − p n − + λ ( g, ω i ) if ρ f (cid:12)(cid:12) G Q p is reducible ,q n − + λ - ε n ( g, ω i ) if ρ f (cid:12)(cid:12) G Q p is irreducible . Proof.
We first note that g is ordinary if and only if ρ f (cid:12)(cid:12) G Q p is reducible (see Remark5.2.1). The corollary then follows from Theorem 5.1, Theorem 4.1, and Lemma3.2. (cid:3) A key lemma.
The main tool in proving Theorem 5.1 is the map α of section4.4. If α ( ϕ f ) = 0, then one can produce a congruence to a weight 2 form, andbegin to compare their Mazur–Tate elements. In this section, we establish thenon-vanishing of α ( ϕ f ) for the forms f we are considering. Lemma 5.4. If f satisfies the hypotheses of Theorem 5.1, then α ( ϕ ± f ) = 0 . Proof.
First note that hypotheses (2) and (3) of Theorem 5.1 imply that k ≡ p − k = 2 is vacuous, and the case k = p + 1 follows from [1,Theorem 3.4(a)]. For k ≥ p , by [1, Theorem 3.4(c)], it suffices to show that ϕ ± f cannot lie in the image of the theta operator θ : H c (Γ , V k − p − (1)) → H c (Γ , V k − ) . We will prove this by showing that no eigensymbol in the image of θ has residualrepresentation isomorphic to ρ f after restriction to I p .Assume first that ρ f | I p is irreducible and thus isomorphic to I (1). For anyweight m ≥
2, let L irr ( m ) denote the set of t ∈ Z / ( p − Z such that there existsan eigenform g on Γ of weight m with ρ g | I p ∼ = I ( t ). ( L irr ( m ) should really beregarded as a subset of the quotient of Z / ( p − Z by the relation that t ∼ pt forall t .) Let L irr θ ( m ) denote the subset of L irr ( m ) of t which occur for forms g in theimage of θ . We aim to show that 1 , p / ∈ L irr θ ( m ) for m < p + 1.By [1, Theorem 3.4] and Lemma 4.10, we have L irr θ ( m ) = ∅ for m ≤ p + 1; L irr θ ( m ) ⊆ (cid:8) t + p + 1 ; t ∈ L irr ( m − p − (cid:9) for m > p + 1; L irr ( m ) ⊆ L irr θ ( m ) ∪ { m ′ + 1 } where m ′ denotes the remainder when m − p −
1. It now follows bya straightforward induction that for k < p + 1, k ≡ p − L irr θ ( k ) = (cid:26) j ( p −
1) + 1 ; 2 ≤ j ≤ k − p − , j = p + 32 (cid:27) . (Note that the induction involves all even k , not just those which are congruent to2 modulo p − p lies in L irr θ ( k ) for such k ; it followsthat ϕ ± f does not lie in the image of θ , as desired.When ρ f | I p is reducible it is necessarily isomorphic to ( ω ∗ ) with ∗ non-zero. Let L red ( m ) ⊆ Z / ( p − Z denote the set of all t such that (cid:0) ω t ∗ ∗ (cid:1) (with ∗ non-zero) canoccur as the restriction to I p of the residual representation of some form of weight k for Γ which has a globally irreducible residual representation. As before, we have L red θ ( m ) = ∅ for m ≤ p + 1; L red θ ( m ) ⊆ (cid:8) t + 1 ; t ∈ L red θ ( m − p − (cid:9) for m > p + 1; L red ( m ) ⊆ L red θ ( m ) ∪ { m − } . Once again, a straightforward induction establishes that for k ≤ p +32 , k ≡ p −
1) we have L red θ ( k ) ⊆ (cid:26) − j ; 0 ≤ j ≤ k − p − − (cid:27) while for p +32 < k < p + 1, k ≡ p −
1) we have L red θ ( k ) ⊆ (cid:26) − j ; 0 ≤ j ≤ k − p − − (cid:27) . In particular, 1 does not lie in L red θ ( k ) for such k , so that ϕ ± f does lie in the imageof θ . (cid:3) AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 21
Proof of Theorem 5.1.
Let ϕ f ∈ H c (Γ , V k − ( O )) denote the modular sym-bol attached to f , and let ϕ f denote its non-zero image in H c (Γ , V k − ). By Lemma5.4, α ( ϕ ± f ) is non-zero. Thus, by Lemma 4.8, µ ± min ( f ) = 0 which establishes thefirst part of the theorem.By Lemma 5.4, α ( ϕ f ) is a (non-zero) eigensymbol in H c (Γ , F ) with the sameHecke-eigenvalues as ϕ f for all primes ℓ (even ℓ = p ). By [1, Proposition 2.5], thereexists an eigenform h ∈ S (Γ ) whose Hecke-eigenvalues reduce to the eigenvaluesof ϕ f . Since ϕ f is p -non-ordinary, the same is true of h . However, this implies that h must be old at p ; indeed, any form of weight 2 which is p -new is automatically p -ordinary. Let g ∈ S (Γ) denote the corresponding eigenform which is new at p ,but has all same Hecke-eigenvalues at primes away from p ; that is, h is in the spanof g ( z ) and g ( pz ).Let ϕ g in H c (Γ , F ) denote the reduction of the modular symbol attached to g . One might except a congruence between ϕ g and α ( ϕ f ). However, the formersymbol has level Γ while the latter has level Γ . If we view ϕ g in H c (Γ , F ), thenit is no longer an eigensymbol at p . Instead, we consider the symbol ϕ g (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) in H c (Γ , F ) which is also an eigensymbol at all primes away from p , and moreover, (cid:0) ϕ g (cid:12)(cid:12) (cid:0) p
00 1 (cid:1)(cid:1) (cid:12)(cid:12) U p = p − X a =0 (cid:0) ϕ g (cid:12)(cid:12) (cid:0) p
00 1 (cid:1)(cid:1) (cid:12)(cid:12) (cid:0) a p (cid:1) = p − X a =0 ϕ g (cid:12)(cid:12) (cid:0) p pa p (cid:1) = p − X a =0 ϕ g (cid:12)(cid:12) ( a )= p − X a =0 ϕ g = p · ϕ g = 0 . Thus, ϕ g (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) is a Hecke-eigensymbol for the full Hecke-algebra. As the same istrue of α ( ϕ f ), by mod p multiplicity one (see [16, Theorem 2]), we have α ( ϕ ± f ) = c ± · ϕ ± g (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) with c ± = 0. Moreover, by changing Ω ± f by a p -unit, we can take c ± equal to 1.Then, by Lemmas 4.6 and 2.6, ϑ n ( f ) = ϑ n ( α ( ϕ f )) = ϑ n ( ϕ g (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) ) = cor nn − (cid:0) ϑ n − ( ϕ g ) (cid:1) completing the proof of theorem.6. Results in small slope
In this section, we will prove a theorem along the lines of Theorem 5.1, butinstead of assuming a bound on the weight of f , we assume on bound on its slope.Interestingly, the proof uses a congruence argument even though the µ -invariantsthat appear need not be zero.6.1. Statement of theorem.Theorem 6.1.
Let f be an eigenform in S k (Γ , Q p ) such that (1) ρ f is irreducible, (2) 0 < ord p ( a p ) < p − , (3) k ( ρ f ) = 2 and ρ f (cid:12)(cid:12) G Q p is not decomposable. Then (1) µ ± min ( f ) ≤ ord p ( a p ) holds for both choices of sign; (2) there exists an eigenform g ∈ S (Γ) with a ℓ ( f ) = a ℓ ( g ) for all primes ℓ = p ,and a choice of cohomological periods Ω f , Ω g ∈ C such that ̟ − a ϑ n,i ( f ) = cor nn − (cid:0) ϑ n − ,i ( g ) (cid:1) in F [ G n ] where a ∈ Z ≥ is such that ord p ( ̟ a ) = µ ε i min ( f ) . We maintain the hypotheses and notation of Theorem 6.1 in the following corol-lary.
Corollary 6.2. If ρ f (cid:12)(cid:12) G Q p is reducible (resp. irreducible), then (1) µ ( θ n,i ( f )) = µ ε i min ( f ) for n ≫ ⇐⇒ µ ( g, ω i ) = 0 (resp. µ ± ( g, ω i ) = 0 ). (2) if the equivalent conditions of (1) hold and n ≫ , then λ ( θ n ( f )) = p n − p n − + λ ( g, ω i ) if ρ f (cid:12)(cid:12) G Q p is reducible ,q n − + λ - ε n ( g, ω i ) if ρ f (cid:12)(cid:12) G Q p is irreducible . Remark 6.3. (1) By results of Buzzard and Gee [4], if k ≡ p −
1) and ord p ( a p ) < ρ f (cid:12)(cid:12) I p ∼ = I (1) and thus hypotheses (1) and (3) are automatic.(2) Hypothesis (2) is necessary as we have found forms of slope p − λ -invariants do not follow the pattern described by Corollary 6.2. In theseexamples, the λ -invariants satisfy λ ( θ n ( f )) = p n − p n − + λ ( g ) if ρ f (cid:12)(cid:12) G Q p is reducible, q n − + λ ε n ( g ) if ρ f (cid:12)(cid:12) G Q p is irreducible,for n ≫ g is some congruent form in weight 2. This phenomenonwill be further explored in section 7.6.2. Filtration lemmas.
Let O be the ring of integers in a finite extension of Q p ,and consider the filtration on V g ( O ) given byFil r ( V g ) = Fil r ( V g ( O )) = g X j =0 b j X j Y g − j ∈ V g ( O ) : p r − j | b j for 0 ≤ j ≤ r − . Recall that the semi-group S ( p ) appearing in the next lemma was introducedin section 4.4. Lemma 6.4.
We have: (1) Fil r ( V g ) is stable under the action of S ( p ) . (2) If P ∈ Fil r ( V g ) , then P (cid:12)(cid:12) (cid:0) a p (cid:1) ∈ p r V g ( O ) .Proof. For (cid:0) a bc d (cid:1) ∈ S ( p ), we have X j Y g − j (cid:12)(cid:12) (cid:0) a bc d (cid:1) = ( dX − cY ) j ( − bX + aY ) g − j . Expanding the above expression and using the fact that p | c and p ∤ a , one seesthat the coefficient of X s Y g − s is divisible by p j − s for s ≤ j . The first part of thelemma follows from this observation. AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 23
For the second part, we have that p r − j X j Y g − j (cid:12)(cid:12) (cid:0) a p (cid:1) = p r − j ( pX ) j ( − aX + Y ) g − j ∈ p r V g ( O )which proves the lemma. (cid:3) Lemma 6.5. If ϕ ∈ H c (Γ , V g ( O )) is a T p -eigensymbol which takes values in Fil r ( V g ) and || ϕ || = 1 , then the slope of ϕ is greater than or equal to r .Proof. Write ϕ (cid:12)(cid:12) T p = λ · ϕ , and choose D ∈ ∆ such that || ϕ ( D ) || = 1. We thenhave(8) λ · ϕ ( D ) = ( ϕ (cid:12)(cid:12) T p )( D ) = p − X a =0 ϕ (cid:0)(cid:0) a p (cid:1) D (cid:1) (cid:12)(cid:12) (cid:0) a p (cid:1) + ϕ (cid:0)(cid:0) p
00 1 (cid:1) D (cid:1) (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) . By Lemma 6.4, all of the terms on the right-hand side are divisible by p r exceptfor possibly the last.To deal with the final term write ϕ (cid:0)(cid:0) p
00 1 (cid:1) D (cid:1) = P gj =0 a j X j Y g − j . By Lemma2.7, we have that g X j =0 a j X j Y g − j (cid:12)(cid:12)(cid:12) (cid:0) − N (cid:1) = g X j =0 a j ( − N Y ) j X g − j = g X j =0 ( − j N a g − j X j Y g − j is also a value of ϕ , and thus is in Fil r ( V g ). In particular, for 0 ≤ j ≤ r , we have p r − j | N a g − j , and hence p r − j | a g − j as gcd( N, p ) = 1. Finally, ϕ (cid:0)(cid:0) p
00 1 (cid:1) D (cid:1) (cid:12)(cid:12)(cid:12) (cid:0) p
00 1 (cid:1) = g X j =0 a j X j Y g − j (cid:12)(cid:12)(cid:12) (cid:0) p
00 1 (cid:1) = g X j =0 a j p g − j X j Y g − j is in p r V g , and thus λ · ϕ ( D ) ∈ p r V g . As || ϕ ( D ) || = 1, we deduce that ord p ( λ ) ≥ r as desired. (cid:3) In what follows, we will need to make use of a finer filtration on V g ( O ). Note thatas an abelian group, Fil r ( V g ) / Fil r +1 ( V g ) is simply ( O /p O ) r +1 . Thus, we introducethe following subfiltration of Fil r ( V g ); for s ≤ r we setFil r,s ( V g ) = g X j =0 b j X j Y g − j ∈ Fil r ( V g ) : p r − j +1 | b j for r + 1 − s ≤ j ≤ r . Note thatFil r ( V g ) = Fil r, ( V g ) ) Fil r, ( V g ) ) · · · ) Fil r,r ( V g ) ) Fil r,r +1 ( V g ) = Fil r +1 ( V g ) . In the following lemma, (cid:0) O /p O ( a j ) (cid:1) ( r ) denotes the S ( p )-module O /p O onwhich γ = (cid:0) a bc d (cid:1) acts by multiplication by det( γ ) r · a j . Lemma 6.6. (1) Fil r,s ( V g ) is stable under the action of S ( p ) . (2) For ≤ s ≤ r , Fil r,s ( V g ) / Fil r,s +1 ( V g ) ∼ = (cid:0) O /p O ( a g − r +2 s ) (cid:1) ( r − s ) as S ( p ) -modules. Moreover, this quotient is generated by the image of themonomial p s X r − s Y g − r + s . Proof.
The first part follows just as in Lemma 6.4. For the second part, directlyfrom the definitions, we have that Fil r,s ( V g ) / Fil r,s +1 ( V g ) is isomorphic to O /p O and is generated by the image of p s X r − s Y g − r + s . For the S ( p )-action, we have p s X r − s Y g − r + s (cid:12)(cid:12) (cid:0) a bc d (cid:1) ≡ p s ( dX ) r − s ( − bX + aY ) g − r + s (mod Fil r,s +1 ( V g )) ≡ d r − s a g − r + s · p s X r − s Y g − r + s (mod Fil r,s +1 ( V g )) . Thus,Fil r,s ( V g ) / Fil r,s +1 ( V g ) ∼ = O /p O ( d r − s a g − r + s ) ∼ = O /p O (( ad ) r − s a g − r +2 s ) ∼ = (cid:0) O /p O (cid:0) a g − r +2 s (cid:1)(cid:1) ( r − s )as desired. (cid:3) The following is a slight refinement of Lemma 6.5, and will be useful in the proofof Theorem 6.1.
Lemma 6.7.
Let ϕ ∈ H c (Γ , V g ( O )) be a T p -eigensymbol which takes values in Fil r,r ( V g ) and such that r ≤ µ min ( ϕ ) < r + 1 . If || ϕ || = 1 , then the slope of ϕ isgreater than or equal to µ min ( ϕ ) .Proof. The proof follows just as in Lemma 6.5. (cid:3)
Proof of Theorem 6.1.
To ease notation, set F a = Fil a ( V k − ( O )), F a,b =Fil a,b ( V k − ( O )), and ϕ = ϕ ± f . Let r ≥ ϕ takes values in F r , and let s ≥ ϕ takes valuesin F r,s . Note that by definition s ≤ r since F r,r +1 = F r +1 .By Lemma 6.5, we have r ≤ ord p ( a p ), and thus hypothesis (2) gives(9) s ≤ r < p − . Our first goal is to show that r = s .Since ϕ does not take all of its values in F r,s +1 , its image in H c (Γ , F r,s /F r,s +1 )is non-zero. Thus, by Lemma 6.6, ϕ gives rise to a non-zero eigensymbol in H c (Γ , O /p O ( a p − − r +2 s ))( r − s );here, we are using that k ≡ p − ̟ t O but not in ̟ t +1 O , projecting modulo ̟ t +1 and dividing by ̟ t gives rise toa non-zero eigensymbol η f ∈ H c (Γ , F ( a p − − r +2 s ))( r − s ) . Then, by [1, Proposition 2.5 and Lemma 2.6], there exists an eigenform g in S (Γ , ω − r +2 s ) such that ρ f ∼ = ρ g ⊗ ω r − s .By Lemma 4.10, we then have one of the following three possibilities:(10) ρ g (cid:12)(cid:12) I p = I (( − r + 2 s ) ′ + 1) , (cid:18) ω − r +2 s +1 ∗ (cid:19) , or (cid:18) ω ∗ ω − r +2 s (cid:19) where, for an integer x , we set x ′ equal to the unique integer j with 0 ≤ j ≤ p − j ≡ x (mod p − ρ f (cid:12)(cid:12) I p = (cid:18) ω − r + s +1 ∗ ω r − s (cid:19) or (cid:18) ω r − s +1 ∗ ω − r + s (cid:19) . Hypothesis (3) implies that ρ f (cid:12)(cid:12) I p ∼ = ( ω ∗ ) with ∗ non-zero, and thus s ≡ r (mod p − s = r as desired. AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 25 If ρ f (cid:12)(cid:12) G Q p is irreducible, we have ρ f (cid:12)(cid:12) G Q p = ( I ( p + ( p − r − s )) if r − s ≤ p − ,I (2 p − p − r − s )) if r − s > p − . By hypothesis (3), we then have ρ f (cid:12)(cid:12) G Q p ∼ = I (1) ∼ = I ( p + ( p − r − s )) or I (2 p − p − r − s )) . In the first case, p + ( p − r − s ) ≡ p (mod p − . Thus, r − s ≡ p or 0 (mod p + 1)which forces s = r . A similar analysis in the second case shows that no such r and s exist. Hence, in all possible cases, r = s and ϕ f takes values in F r,r .By Lemma 6.6, p r Y k − generates F r,r /F r,r +1 , and thus the image of ϕ in H c (Γ , F r,r /F r,r +1 ) ∼ = H c (Γ , O /p O )is given by D (cid:18) p r ϕ ( D ) (cid:12)(cid:12) ( X,Y )=(0 , (cid:19) (mod p ) . This implies that η f is given by D (cid:18) ̟ t p r ϕ ( D ) (cid:12)(cid:12) ( X,Y )=(0 , (cid:19) (mod ̟ ) . By construction, ord p ( ̟ t p r ) = µ ± min ( f ); thus, if we let a be the integer such thatord p ( ̟ a ) = µ ± min ( f ), scaling by a unit then yields the eigensymbol D ̟ a ϕ ( D ) (cid:12)(cid:12) ( X,Y )=(0 , ∈ F in H c (Γ , F ) whose system of Hecke-eigenvalues is the reduction of the system ofeigenvalues attached to f .The argument now proceeds as in Theorem 5.1 to show that ̟ − a θ n,i ( f ) = cor nn − ( θ n − ,i ( g )) in F [ G n ]as desired.Lastly, the inequality µ ± min ( f ) ≤ ord p ( a p ) follows from Lemma 6.7.7. A strange example
In this section, we describe a strange behavior of Iwasawa invariants of formswhich do not satisfy the hypotheses of Theorems 5.1 and 6.1.Take p = 3, and consider the space of cuspforms S (Γ (11) , Q p ). In this space,there are exactly 2 (Galois conjugacy classes of) eigenforms of slope 2 whose residualrepresentations are isomorphic to the 3-torsion on X (11). Let f and f denoterepresentatives from the each of these classes. Note that the associated residualrepresentation is locally reducible at 3 as X (11) is ordinary at 3. Neither Theorem5.1 nor Theorem 6.1 apply directly to these forms as the weight k = 18 is greaterthan p , and the slope 2 is not less than p − Let O j denote the ring of integers of the field generated by the coefficients of f j ,and let ̟ j denote a uniformizer. A computer computation shows that µ +min ( f j ) = 2for j = 1 ,
2, and so we consider the map V ( O j ) −→ O j /p ̟ j O j P ( X, Y ) P (0 ,
1) (mod p ̟ j ) . As this map is Γ ( p )-equivariant, it induces a Hecke-equivariant map α : H c (Γ , V ( O j )) −→ H c (Γ ( p N ) , O j /p ̟ j O j ) . By construction, α ( ϕ + f j ) is non-zero and takes values in p O j /p ̟ j O ∼ = F . Ifwe view α ( ϕ + f j ) in H c (Γ ( p N ) , F ), then it is an eigensymbol whose system ofHecke-eigenvalues is the reduction of the system attached to f j .A computer computation then shows that the subspace of H c (Γ ( p N ) , F ) + with this system of Hecke-eigenvalues is 3-dimensional and generated by ϕ + g (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) , ϕ + g (cid:12)(cid:12) (cid:16) p
00 1 (cid:17) , and ϕ + g (cid:12)(cid:12) (cid:16) p
00 1 (cid:17) where g is the unique normalized eigenform in S (Γ (11)). (Note that mod p multiplicity one is failing for trivial reasons!) Thus, we have α ( ϕ + f j ) = a j, · ϕ + g (cid:12)(cid:12) (cid:0) p
00 1 (cid:1) + a j, · ϕ + g (cid:12)(cid:12) (cid:16) p
00 1 (cid:17) + a j, · ϕ + g (cid:12)(cid:12) (cid:16) p
00 1 (cid:17) , and, in particular,(12) θ n ( f j ) p = a j, · cor nn − ( θ n − ( g ))+ a j, · cor nn − ( θ n − ( g ))+ a j, · cor nn − ( θ n − ( g )) . These equations should allow us to determine the Iwasawa invariants of f j in termsof the invariants of the p -ordinary form g ; in this case, one computes that µ ( g ) = λ ( g ) = 0.A key difference now emerges between f and f ; namely, a computer computa-tion shows that a , = 0 while a , = 0 . This vanishing is significant because for j = 1, the first term on the right hand sideof (12) dominates in calculating λ , and we have λ ( θ n ( f )) = p n − p n − + λ ( g ) = p n − p n − . For j = 2, the second term in (12) dominates and we have λ ( θ n ( f )) = p n − p n − + λ ( g ) = p n − p n − . Thus, there is a “second-order” difference in the rate of growth of the λ -invariantsof f and f .Similar examples exist in the locally irreducible case. For instance, for p = 3,there is an eigenform f in S (Γ (17) , Q p ) whose residual representation is iso-morphic to the 3-torsion in X (17) (which is locally irreducible at 3 as X (17) issupersingular at 3), whose slope is 5, and for which we have λ ( θ n ( f )) = p n − p n − + q n − as opposed to the λ -invariants q n +1 = p n − p n − + q n − which occur in Theorems5.1 and 6.1. AZUR–TATE ELEMENTS OF NON-ORDINARY MODULAR FORMS 27
References [1] Avner Ash and Glenn Stevens,
Modular forms in characteristic ℓ and special values of their L -functions , Duke Math. J. (1986), no. 3, 849–868.[2] , Cohomology of arithmetic groups and congruences between systems of Hecke eigen-values , J. Reine Angew. Math. (1986), 192–220.[3] Laurent Berger,
Repr´esentations modulaires de GL ( Q p ) et repr´esentations galoisiennes dedimension 2 , to appear in Ast´erisque.[4] Kevin Buzzard and Toby Gee, Explicit reduction modulo p of certain crystalline representa-tions , arXiv:0804.1164.[5] Matthew Emerton, Robert Pollack and Tom Weston, Variation of the Iwasawa invariants inHida families , Invent. Math. (2006), no. 3, 523–580.[6] Edixhoven – weight in Serre’s conjecture[7] Ralph Greenberg,
Iwasawa theory for elliptic curves , in Arithmetic theory of elliptic curves(Cetraro, 1997), 51–144, Lecture Notes in Math., 1716, Springer, Berlin, 1999.[8] Ralph Greenberg and Vinayak Vatsal,
On the Iwasawa invariants of elliptic curves , Invent.Math. (2000), no. 1, 17–63.[9] Ralph Greenberg, Adrian Iovita, Robert Pollack,
On the Iwasawa invariants of modularforms at supersingular primes , in preparation.[10] Benedict Gross,
A tameness criterion for Galois representations associated to modular forms(mod p ) , Duke Math. J. (1990), no. 2, 445–517.[11] Masato Kurihara, On the Tate Shafarevich groups over cyclotomic fields of an elliptic curvewith supersingular reduction I , Invent. Math. (2002), no. 1, 195–224.[12] Barry Mazur, John Tate and Jeremy Teitelbaum, On p -adic analogues of the conjectures ofBirch and Swinnerton-Dyer , Invent. Math. (1986), no. 1, 1–48.[13] Bernadette Perrin-Riou, Arithm´etique des courbes elliptiques `a r´eduction supersingulire en p , Experiment. Math. 12 (2003), no. 2, 155–186.[14] Robert Pollack, An algebraic version of a theorem of Kurihara , J. Number Theory 110 (2005),no. 1, 164–177.[15] ,
On the p -adic L -function of a modular form at a supersingular prime , Duke Math.J. 118 (2003), no. 3, 523–558.[16] Kenneth Ribet, Multiplicities of p -finite mod p Galois representations in J ( Np ), Bol. Soc.Brasil. Mat. (N.S.) (1991), no. 2, 177–188.[17] Kenneth Ribet, Congruence relations between modular forms , Proceedings of the Interna-tional Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 503–514.[18] David Rohrlich, On L -functions of elliptic curves and cyclotomic towers , Invent. Math. (1984), no. 3, 409–423.[19] Goro Shimura, Introduction to the arithmetic theory of automorphic functions , PrincetonUniversity Press, 1971.[20] Andrew Wiles,
On ordinary λ -adic representations associated to modular forms , Invent.Math. (1988), 529–573.(Robert Pollack) Department of Mathematics, Boston University, Boston, MA
E-mail address , Robert Pollack: [email protected] (Tom Weston)
Dept. of Mathematics, University of Massachusetts, Amherst, MA
E-mail address , Tom Weston:, Tom Weston: