aa r X i v : . [ m a t h . N T ] S e p THE p -ADIC GROSS–ZAGIER FORMULAON SHIMURA CURVES by Daniel Disegni
Abstract . —
We prove a general formula for the p -adic heights of Heegner points on modular abelianvarieties with potentially ordinary (good or semistable) reduction at the primes above p . The formulais in terms of the cyclotomic derivative of a Rankin–Selberg p -adic L -function, which we construct.It generalises previous work of Perrin-Riou, Howard, and the author, to the context of the work ofYuan–Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods.We further construct analytic functions interpolating Heegner points in the anticyclotomic variables,and obtain a version of our formula for them. It is complemented, when the relevant root number is +1 rather than − , by an anticyclotomic version of the Waldspurger formula.When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies onedivisibility in a p -adic Birch and Swinnerton–Dyer conjecture in anticyclotomic families. Otherapplications described in the text will appear separately. Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1. Heegner points and multiplicity one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. The p -adic L -function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3. p -adic Gross–Zagier formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4. Anticyclotomic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6. History and related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7. Outline of proofs and organisation of the paper . . . . . . . . . . . . . . . . . . . . . . 141.8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.9. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162. p -adic modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1. Modular forms and their q -expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2. Hecke algebra and operators U v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3. Universal Kirillov and Whittaker models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4. p -crtical forms and the p -adic Petersson product . . . . . . . . . . . . . . . . . . . . 243. The p -adic L -function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1. Weil representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2. Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3. Eisenstein family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4. Analytic kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5. Waldspurger’s Rankin–Selberg integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6. Interpolation of local zeta integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.7. Definition and interpolation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 DANIEL DISEGNI p -adic heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1. Local and global height pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2. Heights and intersections on curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3. Integrality crieria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425. Generating series and strategy of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.1. Shimizu’s theta lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2. Hecke correspondences and generating series . . . . . . . . . . . . . . . . . . . . . . . . 455.3. Geometric kernel function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.4. Arithmetic theta lifting and kernel identity . . . . . . . . . . . . . . . . . . . . . . . . . . 496. Local assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.1. Assumptions away from p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2. Assumptions at p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527. Derivative of the analytic kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.1. Whittaker functions for the Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . 537.2. Decomposition of the derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.3. Main result on the derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548. Decomposition of the geometric kernel and comparison . . . . . . . . . . . . . . . . . . . . . . 568.1. Vanishing of the contribution of the Hodge classes . . . . . . . . . . . . . . . . . . 568.2. Decompositon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.3. Comparison of kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579. Local heights at p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599.1. Local Hecke and Galois actions on CM points . . . . . . . . . . . . . . . . . . . . . . . . 599.2. Annihilation of local heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210. Formulas in anticyclotomic families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6510.1. A local construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6610.2. Gross–Zagier and Waldspurger formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6810.3. Birch and Swinnerton-Dyer formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Appendix A. Local integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72A.1. Basic integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72A.2. Interpolation factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72A.3. Toric period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
1. Introduction
The main results of this paper are the general formula for the p -adic heights of Heegner pointsof Theorem B below, and its version in anticyclotomic families (contained in Theorem C). Theyare preceded by a flexible construction of the relevant p -adic L -function (Theorem A), and comple-mented by a version of the Waldspurger formula in anticyclotomic families (presented in Theorem Cas well). In Theorem D, we give an application to a version of the p -adic Birch and Swinnerton-Dyerconjecture in anticyclotomic families. In Theorem E, we state a result on the generic non-vanishingof p -adic heights on CM abelian varieties, as a special case of a theorem to appear in joint workwith A. Burungale.Our theorems are key ingredients of a new Gross–Zagier formula for exceptional zeros [20], andof a universal p -adic Gross–Zagier formula specialising to analogues of Theorem B in all weights.These will be given in separate works. Here we would just like to mention that all of them, as wellas Theorem E, make essential use of the new generality of the present work. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES The rest of this introductory section contains the statements of our results, followed by anoutline of their proofs. To avoid interrupting the flow of exposition, the discussion of previous andrelated works (notably by Perrin-Riou and Howard) has mostly been concentrated in §1.6.
Let A be a simple abelian variety of GL -type over a totally real field F ; recall that this means that M := End ( A ) is a field of dimensionequal to the dimension of A . One knows how to systematically construct points on A when A admits paramtetrisations by Shimura curves in the following sense. Let B be a quaternion algebraover the adèle ring A = A F of F , and assume that B is incoherent , i.e. that its ramificationset Σ B has odd cardinalty. We further assume that Σ B contains all the archimedean places of F . Under these conditions there is a tower of Shimura curves { X U } over F indexed by the opencompact subgroups U ⊂ B ∞× ; let X = X ( B ) := lim ←− U X U . For each U , there is a canonical Hodgeclass ξ U ∈ Pic ( X U ) Q having degree in each connected component, inducing a compatible family ι ξ = ( ι ξ,U ) U of quasi-embeddings (1) ι ξ,U : X U ֒ → J U := Alb X U . We write J := lim ←− J U . The M -vector space π = π A = π A ( B ) := lim −→ U Hom ( J U , A ) is either zero or a smooth irreducible admissible representation of B ∞× . It comes with a naturalstable lattice π Z ⊂ π , and its central character ω A : F × \ A × → M × corresponds, up to twist by the cyclotomic character, to the determinant of the Tate module underthe class field theory isomorphism. When π A is nonzero, A is said to be parametrised by X ( B ) .Under the conditions we are going to impose on A , the existence of such a parametrisation, for asuitable choice of B (see below), is equivalent to the modularity conjecture. Recall that the latterasserts the existence of a unique M -rational (Definition 1.2.1 below) automorphic representation σ A of weight such that there is an equality of L -functions L ( A, s + 1 /
2) = L ( s, σ A ) . Theconjecture is known to be true for “almost all” elliptic curves A (see [34]), and when A F hascomplex multiplication. Heegner points . — Let A be parametrised by X ( B ) and let E be a CM extension of F admitting an A ∞ -embedding E A ∞ ֒ → B ∞ , which we fix; we denote by η the associated quadratic character andby D E its absolute discriminant. Then E × acts on X and by the theory of complex multiplicationeach closed point of the subscheme X E × is defined over E ab , the maximal abelian extension of E .We fix one such CM point P . Let L ( χ ) be a field extension of M and let χ : E × \ E × A ∞ → L ( χ ) × be a finite order Hecke character such that ω A · χ | A ∞ , × = 1; We can view χ as a character of G E := Gal( E/E ) via the reciprocity map of class field theory(normalised, in this work, by sending uniformisers to geometric Frobenii). For each f ∈ π A , wethen have a Heegner point P ( f, χ ) = Z Gal( E ab /E ) f ( ι ξ ( P ) σ ) ⊗ χ ( σ ) dσ ∈ A ( χ ) . Here the integration uses the Haar measure of total volume , and A ( χ ) := ( A ( E ab ) ⊗ M L ( χ ) χ ) Gal( E ab /E ) , (1) By ‘quasi-embedding’, we mean an element of
Hom ( X U , J U ) ⊗ Q , a multiple of which is an embedding. DANIEL DISEGNI where L ( χ ) χ denotes the one-dimensional Galois module L ( χ ) with action given by χ . The func-tional f P ( f, χ ) defines an element of Hom E × A ∞ ( π ⊗ χ, L ( χ )) ⊗ L ( χ ) A ( χ ) . A foundational local result of Tunnell and Saito [48, 54] asserts that, for any irreducible represen-tation π of B × , the L ( χ ) -dimension of H( π, χ ) = Hom E × A ∞ ( π ⊗ χ, L ( χ )) is either zero or one. It is one exactly when, for all places v of F , the local condition ε (1 / , π E,v ⊗ χ v ) = χ v ( − η v ( − ε ( B v ) (1.1.1)holds, where π E is the base-change of π to E , η = η E/F is the quadratic character of A × associatedto E and ε ( B v ) = +1 if B v is split and − if B v is ramified. In this case, denoting by π ∨ the M -contragredient representation, there is an explicit generator Q = Y v ∤ ∞ Q v ∈ H( π, χ ) ⊗ L ( χ ) H( π ∨ , χ − ) defined by integration of local matrix coefficients Q v ( f ,v , f ,v , χ ) = L (1 , η v ) L (1 , π v , ad) ζ F,v (2) L (1 / , π E,v ⊗ χ v ) Z E × v /F × v χ v ( t v )( π ( t v ) f ,v , f ,v ) v dt v , (1.1.2)for a decomposition ( · , · ) = ⊗ v ( · , · ) v of the pairing π ⊗ M π ∨ → M , and Haar measures dt v assigningto O × E,v / O × F v the volume if v is unramified in E and if v ramifies in E . The normalisation issuch that given f , f , all but finitely many terms in the product are equal to . The pairings Q v in fact depend on the choice of decomposition, which in general needs an extension of scalars; theglobal pairing is defined over M and independent of choices.Note that the local root numbers are unchanged if one replaces π by its Jacquet–Langlandstransfer to another quaternion algebra, and that when π = π A they equal the local root numbers ε ( A E,v, , χ v ) of the motive H ( A × Spec F Spec E ) ⊗ M χ [25]. This way one can view the localconditions ε ( A E,v , χ v ) = χ v ( − η v ( − ε ( B v ) as determining a unique totally definite quaternion algebra B ⊃ E A over A , which is incoherentprecisely when the global root number ε ( A E , χ ) = − . In this case, A is parametrised by X ( B ) inthe sense described above if and only if A is modular in the sense that the Galois representationafforded by its Tate module is attached to a cuspidal automorphic representation of GL ( A F ) ofparallel weight . We assume this to be the case. Gross–Zagier formulas . — There is a natural identification π ∨ = π A ∨ , where A ∨ is the dualabelian variety (explicitly, this is induced by the perfect M = End ( A ) -valued pairing f ,U ⊗ f ,U vol( X U ) − f ,U ◦ f ∨ using the canonical autoduality of J U for any sufficiently small U ; thenormalising factor vol( X U ) ∈ Q × is the hyperbolic volume of X U ( C τ ) for any τ : F ֒ → C , see [60,§1.2.2]). Similarly to the above, we have a Heegner point functional P ∨ ( · , χ − ) ∈ H( π ∨ , χ − ) ⊗ L A ∨ ( χ − ) . Then the multiplicity one result of Tunnell and Saito implies that for each bilinearpairing h , i : A ( χ ) ⊗ L ( χ ) A ∨ ( χ − ) → V with values in an L ( χ ) -vector space V , there is an element L ∈ V such that h P ( f , χ ) , P ( f , χ − ) i = L · Q ( f , f , χ ) for all f ∈ π , f ∈ π ∨ . HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES In this framework, we may call “Gross–Zagier formula” a formula for L in terms of L -functions.When h , i is the Néron–Tate height pairing valued in C ι ← ֓ M for an archimedean place ι , thegeneralisation by Yuan–Zhang–Zhang [60] of the classical Gross–Zagier formula ([24, 61–63]) yields L = c E · π F : Q ] | D F | / L ′ (1 / , σ ιA,E ⊗ χ ι )2 L (1 , η ) L (1 , σ ιA , ad) (1.1.3)where c E := ζ F (2)( π/ [ F : Q ] | D E | / L (1 , η ) ∈ Q × , (1.1.4)and, in the present introduction, L -functions are as usual Euler products over all the finite places. (2) (However in the main body of the paper we will embrace the convention of [60] of includingthe archimedean factors.) The most important factor is the central derivative of the L -function L ( s, σ ιA,E ⊗ χ ) .When h , i is the product of the v -adic logarithms on A ( F v ) and A ∨ ( F v ) , for a prime v of F whichsplits in E , the v -adic Waldspurger formula of Liu–Zhang–Zhang [35] (generalising [6]) identifies L with the special value of a v -adic Rankin–Selberg L -function obtained by interpolating the values L (1 / , σ A,E ⊗ χ ′′ ) at anticyclotomic Hecke characters χ ′′ of E of higher weight at v (in particular,the central value for the given character χ lies outside the range of interpolation).The object of this paper is a formula for L when h , i is a p -adic height pairing. In this case L is given by the central derivative of a p -adic Rankin–Selberg L -function obtaining by interpolationof L (1 / , σ A,E , χ ′ ) at finite order Hecke characters of E , precisely up to the factor c E / of (1.1.3).We describe in more detail the objects involved. p -adic L -function. — We construct the relevant p -adic L -function as a function ona space of p -adic characters (which can be regarded as an abelian eigenvariety), characterisedby an interpolation property at locally constant characters. It further depends on a choice oflocal models at p (in the present case, additive characters); this point is relevant for the study offields of rationality and does not seem to have received much attention in the literature on p -adic L -functions. Definition 1.2.1 . — An M -rational (3) cuspidal automorphic representation of GL of weight is a representation σ ∞ of GL ( A ∞ ) on a rational vector space V σ ∞ with End GL ( A ∞ ) σ ∞ = M (then V σ ∞ acquires the structure of an M -vector space), such that σ ∞ ⊗ Q σ (2) ∞ = ⊕ ι : M֒ → C σ ι is adirect sum of irreducible cuspidal automorphic representations; here σ (2) ∞ , a complex representationof GL ( F ∞ ) ∼ = GL ( R ) [ F : Q ] , is the product of discrete series of parallel weight and trivial centralcharacter.We fix from now on a rational prime p . Definition 1.2.2 . — Let F v and L be finite extensions of Q p , let σ v be a smooth irreduciblerepresentation of GL ( F v ) on an L -vector space, and let α v : F × v → O × L be a smooth charactervalued in the units of L . We say that σ v is nearly ordinary for weight with unit character α v if σ v is an infinite-dimensional subrepresentation of the un-normalised principal series Ind( | · | v α v , β v ) (2) In [60], the formula has a slightly different appearance from (1.1.3), owing to the following conventions adoptedthere: the L - and zeta functions are complete including the archimedean factors; the functional Q includesarchimedean factors Q v ( f ,v , f ,v , χ ) , which can be shown to equal to /π ; and finally the product Haar mea-sure on E × A ∞ / A ∞ , × equals | D E | − / times our measure (cf. [60, §1.6.1]).(When “ π ” appears as a factor in a numerical formula, it denotes π = 3 . ... ; there should be no risk of confusionwith the representation π A .) (3) See [60, §3.2.2] for more details on this notion.
DANIEL DISEGNI for some other character β v : F × v → L × . (Concretely, σ v is then either an irreducible principalseries or special of the form St( α v ) := St ⊗ ( α v ◦ det) , where St is the Steinberg representation.)If M is a number field, p is a prime of M above p , and σ v is a representation of GL ( F v ) on an M -vector space, we say that σ v is nearly p -ordinary for weight if there is a finite extension L of M p such that σ v ⊗ M L is.In the rest of this paper we omit the clause ‘for weight ’. (4) Fix an M -rational cuspidal automorphic representation σ ∞ of GL ( A ∞ ) of weight ; if there isno risk of confusion we will lighten the notation and write σ instead of σ ∞ . Let ω : F × \ A × → M × be the central character of σ , which is necessarily of finite order.Fix moreover a prime p of M above p and assume that for all v | p the local components σ v of σ are nearly p -ordinary with respective characters α v . We replace L by its subfield M p ( α ) generatedby the values of all the α v , and we similarly let M ( α ) ⊂ L be the finite extension of M generatedby the values of all the α v . Spaces of p -adic and locally constant characters . — Fix throughout this work an arbitrary compactopen subgroup V p ⊂ b O p, × E := Q w ∤ p O × E,w . Let
Γ = E × A ∞ /E × V p , Γ F = A ∞ , × /F × b O p, × F . Then we have rigid spaces Y ′ = Y ′ ω ( V p ) , Y = Y ω ( V p ) , Y F of respective dimensions [ F : Q ]+1+ δ , [ F : Q ] , δ (where δ ≥ is the Leopoldt defect of F , conjectured to be zero) representing thefunctors on L -affinoid algebras Y ′ ω ( V p )( A ) = { χ ′ : Γ → A × : ω · χ ′ | b O p, × F = 1 } , Y ω ( V p )( A ) = { χ : Γ → A × : ω · χ | A ∞ , × = 1 } , Y F ( A ) = { χ F : Γ F → A × } , where the sets on the right-hand sides consist of continuous homomorphisms. The inclusion Y ⊂ Y ′ sits in the Cartesian diagram(1.2.1) Y / / (cid:15) (cid:15) Y ′ (cid:15) (cid:15) { } / / Y F , where the vertical maps are given by χ ′ χ F = ω · χ ′ | A ∞ , × . When ω = , Y is a group object(the “Cartier dual” of Γ / Γ F ); in general, Y ω is a principal homogeneous space for Y under theaction χ · χ = χ χ .Let µ Q denote the ind-scheme over Q of all roots of unity and µ M its base-change to M .Then there are ind-schemes Y ′ l . c . , Y l . c . , Y l . c .F , ind-finite over M , representing the functors on M -algebras Y ′ l . c . ( A ) = { χ ′ : Γ → µ M ( A ) : ω · χ ′ | b O p, × F = 1 } , Y l . c . ( A ) = { χ : Γ → µ M ( A ) : ω · χ | A ∞ , × = 1 } , Y l . c .F ( A ) = { χ F : Γ F → µ M ( A ) } (characters which are locally constant, or equivalently of finite order). (4) Which we have introduced in order to avoid misleading the reader into thinking of ordinariness of an automorphicrepresentation as a purely local notion (but see [21] for how to approach it as such). HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES Definition 1.2.3 . — Let Y ? be one of the above rigid spaces and Y ? , l . c ., an ⊂ Y ? be the (ind-)rigid space which is the analytification of Y ? , l . c .L := Y ? , l . c . × Spec M Spec L . For any finite extension M ′ of M contained in L , there is a natural map of locally M ′ -ringed spaces j M ′ : Y ? , l . c ., an → Y ? , l . c .M ′ .Let M ′ be a finite extension of M contained in L . We say that a section G of the structure sheafof Y ? is algebraic on Y ? , l . c .M ′ if its restriction to Y ? , l . c ., an equals j ♯M ′ G ′ for a (necessarily unique)section G ′ of the structure sheaf of Y ? , l . c .M ′ . (5) In the situation of the definition, we will abusively still denote by G the function G ′ on Y ? , l . c .M ′ . Local additive character . — Let v be a non-archimedean place of F , p v ⊂ O F,v the maximal ideal, d v ⊂ O F,v the different. We define the space of additive characters of F v of level to be Ψ v := Hom ( F v /d − v O F,v , µ Q ) − Hom ( F v /p − v d − v O F,v , µ Q ) , where we regard Hom ( F v /p nv O F,v , µ Q ) as a profinite group scheme over Q . (6) The scheme Ψ v isa torsor for the action of O × F,v (viewed as a constant profinite group scheme over Q ) by a.ψ ( x ) := ψ ( ax ) .If ω ′ v : O × F,v → O ( X ) × is a continuous character for a scheme or rigid space X , we denoteby O X × Ψ v ( ω ′ v ) ⊂ O X × Ψ v the subsheaf of functions G satisfying G ( x, a.ψ ) = ω ′ v ( a )( x ) G ( x, ψ ) for a ∈ O × F,v . By the defining property we can identify O X × Ψ v ( ω ′ v ) with p X ∗ O X × Ψ v ( ω ′ v ) (where p X : X × Ψ v → X is the projection), a locally free rank one O X -module with action by G Q :=Gal( Q / Q ) . Finally we denote Ψ p := Q v | p Ψ v and, if ω ′ p = Q v | p ω ′ v : O × F,p → O ( X ) × , O X × Ψ p ( ω ′ p ) = O v | p O X × Ψ v ( ω ′ v ) , where the tensor product is in the category of O X -modules. Its space of global sections over X will be denoted by O X × Ψ p ( X × Ψ p , ω ′ p ) or simply O X × Ψ p ( X , ω ′ p ) .These sheaves will appear in the next theorem with ω ′ v = ω v χ − F, univ ,v : O × F,v → O ( Y ′ ) × where ω v is the central character of σ v and χ F, univ ,v : O × F,v → O ( Y F ) × → O ( Y ′ ) × comes from the restrictionof the universal O ( Y F ) × -valued character of Γ F . As Ψ p is a scheme over Q and χ F, univ is obviouslyalgebraic on Y l . c .F , the notion of Definition 1.2.3 extends to define Y ′ l . c .M ′ -algebraicity of sections of O Y ′ × Ψ p ( ω ′ v ) .As a last preliminary, we introduce notation for bounded functions: if X is a rigid space then O X ( X ) b ⊂ O X ( X ) is the space of global sections G such that that sup x ∈ X | G ( x ) | is finite;similarly, in the above situation, we let O X × Ψ p ( X , ω ′ p ) b := { G ∈ O X × Ψ p ( X , ω ′ p ) : sup x ∈ X | G ( x, ψ ) | is finite for some ψ ∈ Ψ p } . As ω ′ p is continuous, ω ′ p ( a ) is bounded in a ∈ O × F,p : we could then equivalently replace “is finite forsome ψ ∈ Ψ p ” with “is uniformly bounded for all ψ ∈ Ψ p ”. Theorem A . —
There is a bounded analytic function L p,α ( σ E ) ∈ O Y ′ × Ψ p ( Y ′ , ω − p χ F, univ ,p ) b , uniquely determined by the following property: L p,α ( σ E ) is algebraic on Y ′ l . c .M ( α ) × Ψ p , and for each C -valued geometric point ( χ ′ , ψ p ) ∈ Y ′ l . c .M ( α ) × Ψ p ( C ) , (5) To avoid all confusions due to the clash of notation, Y l . c .F will always denote the M -scheme of locally constantcharacters of Γ F introduced above, and not the ‘base-change of Y l . c . to F ’ (which is not defined as M is not asubfield of F in the generality adopted here). (6) If F v = Q p , then Hom ( F v / O F,v , µ Q ) = T p µ Q , the p -adic Tate module of roots of unity. One could also constructand use a scheme parametrising all nontrivial characters of F v . DANIEL DISEGNI letting ι : M ( α ) ֒ → C be the embedding induced by the composition Spec C χ ′ → Y ′ l . c .M ( α ) → Spec M ( α ) ,we have L p,α ( σ E )( χ ′ , ψ p ) = Y v | p Z ◦ v ( χ ′ v , ψ v ) π F : Q ] | D F | / L (1 / , σ ιE ⊗ χ ′ )2 L (1 , η ) L (1 , σ ι , ad) in C . The interpolation factor is explicitly Z ◦ v ( χ ′ v , ψ v ) := ζ F,v (2) L (1 , η v ) L (1 / , σ E,v ⊗ χ ′ v ) Y w | v Z w ( χ ′ w , ψ v ) with Z w ( χ ′ w , ψ v ) = α v ( ̟ v ) − v ( D ) χ ′ w ( ̟ w ) − v ( D ) − α v ( ̟ v ) − f w χ ′ w ( ̟ w ) − − α v ( ̟ v ) f w χ ′ w ( ̟ w ) q − f w F,v if χ ′ w · α v ◦ q w is unramified, τ ( χ ′ w · α v ◦ q, ψ E w ) if χ ′ w · α v ◦ q w is ramified.Here d , D ∈ A ∞ , × are generators of the different of F and the relative discriminant of E/F respectively, q w is the relative norm of E/F , f w is the inertia degree of w | v , and q F,v is thecardinality of the residue field at v ; finally, for any character e χ ′ w of E × w of conductor f , τ ( e χ ′ w , ψ E w ) := Z w ( t )= − w ( f ) e χ ′ w ( t ) ψ E,w ( t ) dt with dt the restriction of the additive Haar measure on E giving vol( O E , dt ) = 1 , and ψ E,w = ψ F,v ◦ Tr E w /F v . Remark 1.2.4 . — It follows from the description of Lemma A.1.1 that the interpolation factors Z ◦ v , Z w are sections of O Y ′ l . c .v × Ψ v ( ω v χ − F, univ ,v ) , where Y ′ l . c .v is the ind-finite reduced ind-schemeover M ( α ) representing µ M ( α ) -valued characters of E × v . (Later, we will also similarly denote by Y l . c .v ⊂ Y ′ l . c .v the subscheme of characters satisfying χ v | F × v = ω − v .)In fact, we only construct L p,α ( σ E ) as a bounded section of O Y ′ × Ψ p ( ω − p χ F, univ ,p )( D ) , where D is a divisor on Y ′ supported away from Y (i.e., for any polynomial function G on Y ′ with divisorof zeroes ≥ D , the function G · L p,α ( σ E ) is a bounded global section of O Y ′ × Ψ p ( ω − p χ F, univ ,p ) ); (7) see Theorem 3.7.1 together with Proposition A.2.2 for the precise statement. This is sufficient forour purposes and to determine L p,α ( σ E ) uniquely. One can then deduce that it is possible to take D = 0 by comparing our p -adic L -function to some other construction where this difficulty doesnot arise. One such construction has been announced by David Hansen. p -adic Gross–Zagier formula. — Let us go back to the situation in which A is a modularabelian variety of GL -type, associated with an automorphic representation σ A of Res F/ Q GL ofcharacter ω = ω A . p -adic heights . — Several authors (Mazur–Tate, Schneider, Zarhin, Nekovář, . . . ) have defined p -adic height pairings on A ( F ) × A ∨ ( F ) for an abelian variety A . These pairings are analogous tothe classical Néron–Tate height pairings: in particular they admit a decomposition into a sum oflocal symbols indexed by the ( finite ) places of F ; for v ∤ p such symbols can be calculated fromintersections of zero-cycles and degree-zero divisors on the local integral models of A .In the general context of Nekovář [39], adopted in this paper and recalled in §4.1, height pairingscan be defined for any geometric Galois representation V over a p -adic field; we are interested in thecase V = V p A ⊗ M p L where M = End A and L is a finite extension of a p -adic completion M p of M . Different from the Néron–Tate heights, p -adic heights are associated with the auxiliary choice (7) A similar difficulty is encountered for example by Hida in [26]. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES of splittings of the Hodge filtration on D dR ( V | G Fv ) for the primes v | p ; in our case, D dR ( V | G Fv ) = H ( A ∨ /F v ) ⊗ M p L . When V | G Fv is potentially ordinary, meaning that it is reducible in thecategory of de Rham representations (see more precisely Definition 4.1.1), (8) there is a canonicalsuch choice. If A is modular corresponding to an M -rational cuspidal automorphic representation σ ∞ A , it follows from [13, Théorème A], together with [41, (proof of) Proposition 12.11.5 (iv)], thatthe restriction of V = V p A ⊗ L to G F v is potentially ordinary if and only if σ A,v ⊗ L is nearly p -ordinary.We assume this to be the case for all v | p . One then has a canonical p -adic height pairing(1.3.1) h , i : A ( F ) Q ⊗ M A ∨ ( F ) Q → Γ F ˆ ⊗ L. whose precise definition will be recalled at the end of §4.1. Its equivariance properties under theaction of G F = Gal( F /F ) allow to deduce from it pairings(1.3.2) h , i : A ( χ ) ⊗ L ( χ ) A ∨ ( χ − ) → Γ F ˆ ⊗ L ( χ ) for any character χ ∈ Y l . c .L . Remark 1.3.1 . — Suppose that ℓ : Γ F → L ( χ ) is any continuous homomorphism such that, forall v | p , ℓ v | O × F,v = 0 ; we then call ℓ a ramified logarithm . Then it is conjectured, but not known ingeneral, that the the pairings deduced from (1.3.2) by composition with ℓ are non-degenerate. SeeTheorem E for a new result in this direction. Remark 1.3.2 . — If χ is not exceptional in the sense of the next definition, then (1.3.2) is knownto coincide with the norm-adapted height pairings à la Schneider [39, 49], by [39], and with theMazur–Tate [36] height pairings, by [31]. Definition 1.3.3 . — A locally constant character χ w of E × w is said to be not exceptional if Z w ( χ w ) = 0 . A character χ ∈ Y l . c .M ( α ) is said to be not exceptional if for all w | p , χ w is notexceptional. The formula . — Let Y = Y ω ⊂ Y ′ = Y ′ ω be the rigid spaces defined above. Denote by I Y ⊂ O Y ′ the ideal sheaf of Y and by N ∗ Y / Y ′ = ( I Y / I Y ) | Y the conormal sheaf. By (1.2.1), it is canonicallytrivial: N ∗ Y / Y ′ ∼ = O Y ⊗ T ∗ Y F ∼ = O Y ⊗ (Γ F ˆ ⊗ L ) . For a section G of I Y , denote d F G ∈ N ∗ Y / Y ′ its image; it can be thought of as the differential inthe δ cyclotomic variable(s).Let χ ∈ Y l . c ., an be a character such that ε ( A E , χ ) = − ; denote by L ( χ ) its residue field.By the interpolation property, the complex functional equation and the constancy of local rootnumbers, the p -adic L -function L p,α ( σ A,E ) is a section of I Y in the connected component of χ ∈ Y ′ (see Lemma 10.2.2). Let B be the incoherent quaternion algebra determined by (1.1.1)and let π A = π A ( B ) , π A ∨ = π A ∨ ( B ) . Theorem B . —
Suppose that for all v | p , A/F v has potentially p -ordinary good or semistable re-duction, E v /F v is split, and χ is not exceptional (Definition 1.3.3). Then for all f ∈ π A , f ∈ π A ∨ we have h P ( f , χ ) , P ∨ ( f , χ − ) i = c E · Y v | p Z ◦ v ( χ v ) − · d F L p,α ( σ A,E )( χ ) · Q ( f , f , χ ) in N ∗ Y / Y ′ | χ ∼ = Γ F ˆ ⊗ L ( χ ) . Here c E is as in (1.1.4) . (8) This is a p -partial version of the notion of A F v acquiring ordinary (good or semistable) reduction over a finiteextension of F v . DANIEL DISEGNI
In the right-hand side, we have considered Remark 1.2.4 and used the canonical isomorphism O Ψ p ( ω − p ) ⊗ M O Ψ p ( ω p ) = M . Consider the setup of §1.2. Recall that in the case ε (1 / , σ E , χ ) =+1 , the definite quaternion algebra B defined by (1.1.1) is coherent, i.e. it arises as B = B ⊗ F A F for a quaternion algebra B over F ; we may assume that the embedding E A ֒ → B arises froman embedding i : E ֒ → B . Let π be the automoprhic representation of B ( A ) × attached to σ bythe Jacquet–Langlands correspondence; it is realised in the space of locally constant functions B × \ B × → M , and this gives a stable lattice π O M ⊂ π . Then, given a character χ ∈ Y l . c . , theformalism of §1.1 applies to the period functional p ∈ H( π, χ ) defined by p ( f, χ ) := Z E × \ E × A ∞ f ( i ( t )) χ ( t ) dt and to its dual p ∨ ( · , χ − ) ∈ H( π ∨ , χ − ) . Here dt is the Haar measure of total volume .The formula expressing the decomposition of their product was proved by Waldspurger (see [56]or [60]): for all finite order characters χ : E × \ E × A → M ( χ ) × valued in some extension M ( χ ) ⊃ M ,and for all f ∈ π , f ∈ π ∨ , we have p ( f , χ ) p ∨ ( f , χ − ) = c E · π F : Q ] | D F | / L (1 / , σ E ⊗ χ )2 L (1 , η ) L (1 , σ, ad) · Q ( f , f , χ ) (1.4.1)in M ( χ ) . Notice that here we could trivially modify the right-hand side to replace the complex L -function with the p -adic L -function, thanks to the interpolation property defining the latter.The L -function terms of both the Waldspurger and the p -adic Gross–Zagier formula thus admitan interpolation as analytic functions (or sections of a sheaf) on Y ω . We can show that the otherterms do as well.Let π be the M -rational representation of the (coherent or incoherent) quaternion algebra B × ⊃ E × A considered above, with central character ω . It will be convenient to denote π + = π , π − = π ∨ , A + = A , A − = A ∨ , P + = P , P − = P ∨ , p + = p , p − = p ∨ , Y ± = Y ω ± .We have a natural isomorphism Y + ∼ = Y − given by inversion. If F is a sheaf on Y − , we denoteby F ι its pullback to a sheaf on Y + ; the same notation is used to transfer sections of such sheaves. Big Selmer groups and heights . — Let χ ± univ : Γ → ( O ( Y ± ) b ) × be the tautological character suchthat χ ± univ ( t )( χ ) = χ ( t ) ± for all χ ∈ Y ± , and define an O ( Y ± ) b -module S p ( A ± E , χ ± univ , Y ± ) b := H f ( E, V p A ± E ⊗ O ( Y ± ) b ( χ ± univ )) , where O ( Y ± ) b ( χ ± univ ) denotes the module of bounded global sections O ( Y ± ) b with G E -action by χ ± univ . Here the subscript f denotes the group of those continuous cohomology classes which areunramified away from p and such that, for each prime w | p of E , the restriction to a decompositiongroup at a prime w is in the kernel of H ( E w , V p A ± E ⊗ O ( Y ± ) b ( χ ± univ )) → H ( E w , V p A ± E | − G E,w ⊗ O ( Y ± ) b ( χ ± univ )) , where V p A ± E | − G E,w is the maximal potentially unramified quotient of V p A ± E | G E,w (cf. §4.1). Forevery non-exceptional χ ± ∈ Y l . c . ± , the specialisation S p ( A E , χ ± univ , Y ± ) b ⊗ L ( χ ) is isomorphic tothe target of the Kummer map κ : A ± ( χ ± ) → H f ( E, V p A ⊗ L ( χ ± ) χ ± ) . The work of Nekovář [41] explains the exceptional specialisations and provides a height pairingon the big Selmer gorups. The key underlying object is the
Selmer complex f RΓ f ( E, V p A ± ⊗ O ( Y ◦± ) b ( χ ± univ )) , (1.4.2) HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES an object in the derived category of O ( Y ± ) b -modules defined as in [41, §0.8] taking T = V p A ± ⊗ O ( Y ◦± ) b ( χ ± univ ) and U + w = V p A ± | + G Ew ⊗ O ( Y ◦± ) b ( χ ± univ ) in the notation of loc. cit. Its first coho-mology group e H f ( E, V p A ± ⊗ O ( Y ◦± ) b ( χ ± univ )) satisfies the following property. For every L -algebra quotient R of O ( Y ◦ + ) b , letting χ ± R : Γ → R × be the character deduced from χ ± univ , there is an exact sequence ( ibid. (0.8.0.1)) → M w | p H ( E w , V p A ± E | − G E,w ⊗ R ( χ ± R )) → e H f ( E, V p A ± ⊗ O ( Y ◦ + ) b ( χ ± univ )) ⊗ R → H f ( E, V p A ± ⊗ R ( χ ± R )) → . (1.4.3)When R = O ( Y ± ) b itself, each group H ( E w , V p A ± E | − G E,w ⊗ O ( Y ± ) b ( χ ± univ )) vanishes as χ univ ,w isinfinitely ramified; hence e H f ( E, V p A ± ⊗ O ( Y ◦± ) b ( χ ± univ )) ∼ = S p ( A ± E , χ ± univ , Y ± ) b . When R = L ( χ ) with χ ∈ Y l . c . , the group H ( E w , V p A ± E | − G E,w ⊗ L ( χ ± ) χ ± ) vanishes unless χ w · α v ◦ q w = on E × w , that is unless χ w is exceptional.Finally, by [41, Chapter 11], there is a big height pairing D , E : S p ( A + E , χ +univ , Y + ) b ⊗ O Y + S p ( A − E , χ − univ , Y − ) b ,ι → N ∗ Y + / Y ′ + ( Y + ) b (1.4.4)interpolating the height pairings on H f ( E, V p A ⊗ L ( χ ± ) χ ± ) for non-exceptional χ ∈ Y l . c . (andmore generally certain ‘extended’ pairings on e H f ( E, V p A ⊗ L ( χ ± ) χ ± ) for all χ ∈ Y l . c . ; these willplay no role here). Theta elements and anticyclotomic formulas . — Keep the assumptions that for all v | p , E v /F v issplit and π v ∼ = σ v is p -nearly ordinary with unit character α v . Then, after tensoring with O Ψ p (Ψ p ) ,we will have a decomposition π ± ∼ = π ± ,p ⊗ π ± p , which is an isometry with respect to pairings ( , ) p , ( , ) p on each of the factors. By (1.1.2), for each χ = χ p χ p ∈ Y ◦ l . c .M we can then define a toricperiod Q p ( f + ,p , f − ,p , χ ) ∈ M ( χ ) ⊗ O Ψ p ( ω − p ) . (1.4.5)Given f ± ,p ∈ π ± ,p , we will construct an explicit pair of elements f ± α = ( f ± α,V p ) = ( f ± ,p ⊗ f α,p,V p ) V p ∈ π ± ,pM ( α ) ⊗ lim ←− V p π ± ,V p p (1.4.6)where the inverse system is indexed by compact open subgroups V p ⊂ E × p ⊂ B × p containing Ker ( ω p ) , with transition maps being given by averages under their π ± p -action. We compute inLemma 10.1.2 that we have Q p ( f + α,p , f − α,p ) = ζ F,p (2) − Y v | p Z ◦ v as sections of N v | p O Y l . c .v × Ψ v ( ω v ) , where the left-hand side in the above expression is computed,for each χ p ∈ Q v | p Y l . c .v , as the limit of Q p ( f + α,p,V p , f − α,p,V p ) as V p → Ker ( ω p ) .For the following theorem, note that all the local signs in (1.1.1) extend to locally constantfunctions of Y + (this is a simple special case of [45, Proposition 3.3.4]); the quaternion algebraover A determined by (1.1.1) is then also constant along the connected components of Y + . Wewill say that a connected component Y ◦ ⊂ Y + is of type ε ∈ {± } if ε (1 / , σ E , χ ) = ε along Y ◦ . Theorem C . —
Let Y ◦ + ⊂ Y + be a connected component of type ε , let B be the quaternionalgebra determined by (1.1.1) , and let π ± be the representations of B × constructed above. Finally,let Y ◦− ⊂ Y − be the image of Y ◦ + under the inversion map. DANIEL DISEGNI
1. (Theta elements.) For each f ± ,p ∈ π ± ,p , there are elements Θ ± α ( f ± ,p ) ∈ O Y + ( Y ◦± ) b if ε = +1 , P ± α ( f ± ,p ) ∈ S p ( A ± E , χ ± univ , Y ◦± ) b if ε = − ,uniquely determined by the property that, for any compact open subgroup V p ⊂ E × p and any V p -invariant character χ ± ∈ Y ◦± , we have Θ ± α ( f ± ,p )( χ ± ) = p ( f ± α,V p , χ ± ) , P ± α ( f ± ,p )( χ ± ) = κ ( P ( f ± α,V p , χ ± )) . where f ± α is the element (1.4.6) .2. There is an element Q = ζ F,p (2) − Y v ∤ p Q v ∈ Hom O ( Y ◦ + ) b [ E × A p ∞ ] ( π + ,p ⊗ π − ,p ⊗ O ( Y ◦ + ) b , O ( Y ◦ + ) b ⊗ O Ψ p ( ω − p )) uniquely determined by the property that, for all f ± ,p ∈ π ± and all χ ∈ Y ◦ l . c . + , we have Q ( f + ,p , f − ,p )( χ ) = ζ F,p (2) − · Q p ( f + ,p , f − ,p , χ p ) .
3. (Anticyclotomic Waldspurger formula.) If ε = +1 , we have Θ + α ( f + ,p ) · Θ − α ( f − ,p ) ι = c E · L p,α ( σ E ) · Q ( f + ,p , f − ,p ) in O ( Y ◦ + ) b .4. (Anticyclotomic Gross–Zagier formula.) If ε = − , we have D P + α ( f + ,p ) , P − α ( f − ,p ) ι E = c E · d F L p,α ( σ E ) · Q ( f + ,p , f − ,p ) in N ∗ Y + / Y ′ + ( Y ◦ + ) b . In parts 3 and 4, we have used the canonical isomorphism O Ψ p ( ω p ) ⊗ O Ψ p ( ω − p ) ∼ = M . Theheight pairing of part 4 is (1.4.4). Remark 1.4.1 . — Theorem C.4 specialises to at an exceptional character χ ∈ Y l . c . , andin fact by the archimedean Gross–Zagier formula of [60] formula it follows that the ‘pair of points’ P + α ( f + ,p ) ⊗ P − α ( f − ,p ) ι itself vanishes there. The leading term of L p,α at exceptional charactersis studied in [20]. Theorem B has by now standard applications to the p -adic and theclassical Birch and Swinnerton–Dyer conjectures; the interested reader will have no difficulty inobtaining them as in [19, 42]. We obtain in particular one p -divisibility in the classical Birch andSwinnerton–Dyer conjecture for a p -ordinary CM elliptic curve A over a totally real field as in[19, Theorem D] without the spurious assumptions of loc. cit. on the behaviour of p in F . In therest of this subsection we describe two other applications. On the p -adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families . — The nexttheorem, which can be thought of as a case of the p -adic Birch and Swinnerton-Dyer conjecturein anticyclotomic families, combines Theorem C.4 with work of Fouquet [22] to generalise a resultof Howard [28] towards a conjecture of Perrin-Riou [43]. We first introduce some notation: let Λ := O ( Y ◦ + ) b , and let the anticyclotomic height-regulator R ⊂ Λ ˆ ⊗ Sym r Γ F (1.5.1) HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES be the discriminant of (1.4.4) on the Λ -module S p ( A + E , χ +univ , Y ◦ + ) b ⊗ Λ S p ( A − E , χ − univ , Y ◦− ) b ,ι , where the integer r in (1.5.1) is the generic rank of the finite type Λ -module S p ( A + E , χ +univ , Y ◦ + ) b . Re-call that this module is the first cohomology of the Selmer complex f RΓ f ( E, V p A + ⊗ O ( Y ◦ + ) b ( χ +univ )) of (1.4.2). Let e H f ( E, V p A + ⊗ O ( Y ) b ( χ univ )) tors be the torsion part of the second cohomology group. Its characteristic ideal in Λ can roughly bethought of as interpolating the p -parts of the rational terms (order of the Tate–Shafarevich group,Tamagawa numbers) appearing on the algebraic side of the Birch and Swinnerton-Dyer conjecturefor A ( χ ) . Theorem D . —
In the situation of Theorem C.4, assume furthermore that – p ≥ ; – V p A is potentially crystalline as a G F v -representation for all v | p ; – the character ω is trivial and Y ◦ is the connected component of ∈ Y ; – the residual representation ρ : G F → Aut F p ( T p A ⊗ F p ) is irreducible (where F p is the residuefield of M p ), and it remains irreducible when restricted to the Galois group of the Hilbertclass field of E ; – for all v | p , the image of ρ | G F,v is not scalar.Then S p ( A E , χ univ , Y ◦ ) b , S p ( A E , χ − , Y ◦ ) b ,ι both have generic rank over Λ , a non-torsion element of their tensor product over Λ is given byany P + α ( f + ,p ) ⊗ P − α ( f − ,p ) ι such that Q ( f + ,p , f − ,p ) = 0 , and (1.5.2) (d F L p,α ( σ E ) | Y ◦ ) ⊂ R · char Λ e H f ( E, V p A ⊗ Λ( χ univ )) tors as Λ -submodules of Λ ˆ ⊗ Γ F . The ‘potentially crystalline’ assumption for V p A , which is satisfied if A has potentially goodreduction at all v | p , is imposed in order for V p A ⊗ O ( Y ◦ ) b to be ‘non-exceptional’ in the senseof [22] (which is more restrictive than ours); the assumption on ω allows to invoke the resultsof [1, 16] on the non-vanishing of anticyclotomic Heegner points, and to write A = A + = A − , Y = Y + = Y − = Y . See [22, Theorem B (ii)] for the exact assumptions needed, which areslightly weaker.The proof of Theorem D will be given in §10.3. Remark 1.5.1 . — When F = Q , the converse divisibility to (1.5.2) was recently proved by X.Wan [58] under some assumptions. Generic non-vanishing of p -adic heights on CM abelian varieties . — The non-vanishing of (cyclo-tomic) p -adic heights is in general, as we have mentioned, a deep conjecture (or a “strong suspicion”)of Schneider [50]. The following result provides some new evidence towards it. It is a corollaryof Thereom C.4 together with the non-vanishing results for Katz p -adic L -functions of Hida [27],Hsieh [29], and Burungale [10] (via a factorisation of the p -adic L -function). The result is a spe-cial case of a finer one to appear in forthcoming joint work with A. Burungale. For CM ellipticcurves over Q , it was known as a consequence of different non-vanishing results of Bertrand [7]and Rohrlich [47] (see [2, Appendix A, by K. Rubin]). DANIEL DISEGNI
Theorem E . —
In the situation of Theorem C.4, suppose that A E has complex multiplication (9) and that p ∤ D F h − E , where h − E = h E /h F is the relative class number. Let (cid:10) , (cid:11) cyc be the pairingdeduced from (1.4.4) by the map N Y / Y ′ ( Y ◦ ) b ∼ = O ( Y ◦ ) b ˆ ⊗ Γ F → O ( Y ◦ ) b ˆ ⊗ Γ cyc , where Γ cyc = Γ Q viewed as a quotient of Γ F via the adèlic norm map.Then for any f ± ,p such that Q ( f + ,p , f − ,p ) = 0 in O ( Y ◦ ) b , we have D P + α ( f + ,p ) , P − α ( f − ,p ) ι E cyc = 0 in O ( Y ◦ ) b ˆ ⊗ Γ cyc . We briefly discuss previous work towards our main theo-rems, and some related works. We will loosely term “classical context” the following specialisationof the setting of our main resutls: A is an elliptic curve over Q with conductor N and good or-dinary reduction at p ; p is odd; the quadratic imaginary field E has discriminant coprime to N and it satisfies the Heegner condition: all primes dividing N split in E (this implies that B ∞ issplit); the parametrisation f : J → A factors through the Jacobian of the modular curve X ( N ) ;the character χ is unramified everywhere, or unramified away from p . Ancestors . — In the classical context, Theorems A and B were proved by Perrin-Riou [42]; anintermediate step towards the present generality was taken in [19]. When
E/F is split above p ,Theorem A can essentially be deduced from a general theorem of Hida [26] (cf. [57, §7.3]), exceptfor the location of the possible poles. Theorem C.4 and Theorem D in the classical context are dueto Howard [28] (in fact Theorems B and C.4 were first envisioned by Mazur [37] in that context,whereas Perrin-Riou [43] had conjectured the equality in (1.5.2)). Theorem C.3 is hardly new andhas many antecedents in the literature: see e.g. [55] and references therein. Relatives . — Some analogues of Theorem B were proven in situations which differ from the clas-sical context in directions which are orthogonal to those of the present work: Nekovář [40] andShnidman [52] dealt with the case of higher weights; Kobayashi [33] dealt with the case of ellipticcurves with supersingular reduction.
Friends . — We have already mentioned two other fully general Gross–Zagier formulas in the senseof §1.1, namely the original archimedean one of [60] generalising [24], and a different p -adic formulaproved in [35] generalising [6]. The panorama of existing formulas of this type is complemented bya handful of results, mostly in the classical context, valid in the presence of an exceptional zero(the case excluded in Theorem B). We refer the reader to [20] where we prove a new such formulafor p -adic heights and review other ones due to Bertolini–Darmon. It is to be expected that all ofthose results should be generalisable to the framework of §1.1. Children . — Finally, explicit versions of any Gross–Zagier formula in the framework of §1.1 canbe obtained by the explicit computation of the local integrals Q v . This is carried out in [12]where it is applied to the cases of the archimedean Gross–Zagier formula and of the Waldspurgerformula; the application to an explicit version of Theorem B can be obtained exactly in the samemanner. An explicit version of the anticyclotomic formulas of Theorem C can also be obtained asa consequence: see [20] for a special case. Let us briefly explain the mainarguments and at the same time the organisation of the paper.For the sake of simplicity, the notation used in this introductory discussion slightly differs fromthat of the body text, and we ignore powers of π , square roots of discriminants, and choices ofadditive character. (9) In the strict sense that the algebra
End ( A E ) of endomorphisms defined over E is a CM field. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES Construction of the p -adic L -function (§3) . — It is crucial for us to have a flexible constructionwhich does not depend on choices of newforms. The starting point is Waldspurger’s [56] Rankin–Selberg integral ( ϕ, I ( φ, χ ′ )) Pet L (1 , σ, ad) /ζ F (2) = L (1 / , σ E ⊗ χ ′ ) L (1 , η ) Y v R ♮v ( ϕ v , φ v , χ ′ v ) , (1.7.1)where ϕ ∈ σ , I ( φ, χ ′ ) is a mixed theta-Eisenstein series depending on a choice of an adèlic Schwartzfunction φ , and R ♮v ( ϕ v , φ v , χ ′ ) are normalised local integrals (almost all of which are equal to ).Then, after dividing both sides by the period L (1 , σ, ad) , we can: – interpolate the kernel χ ′ I (0 , φ, χ ′ ) to a Y ′ -family I ( φ p ∞ ; χ ′ ) of p -adic modular formsfor any choice of the components φ p ∞ , and a well-chosen φ p ∞ (we will set φ v ( x, u ) to be“standard” at v |∞ , and close to a delta function in x at v | p ); – interpolate the functional “Petersson product with ϕ ” to a functional ℓ ϕ p ,α on p -adic modularforms, for any ϕ ∈ σ which is a ‘ U v -eigenvector of eigenvalue α v ’ at the places v | p , and isantiholomorphic at infinity; – interpolate the normalised local integrals χ ′ R ♮v ( ϕ v , φ v , χ ′ v ) to functions R ♮v ( ϕ v , φ v ) for all v ∤ p ∞ and any ϕ v , φ v .To conclude, we cover Y ′ by finitely many open subsets U i ; for each i we choose appropriate ϕ p , φ p and we define (10) L p,α ( σ E ) | U i = ℓ ϕ p ,α ( I ( ϕ p ∞ )) Q v ∤ p ∞ R ♮v ( ϕ v , φ v ) . The explicit computation of the local integrals at p (in the Appendix) and at infinity yields theinterpolation factor. Proof of the Gross–Zagier formula and its anticyclotomic version . — We outline the main argu-ments of our proof, with an emphasis on the reduction steps.
Multiplicity one . — We borrow or adapt many ideas (and calculations) from [60], in particular thesystematic use of the multiplicity one principle of §1.1. As both sides of the formula are functionalsin the same one-dimensional vector space, it is enough to prove the result for one pair f , f with Q ( f , f , χ ) = 0 ; finding such f ⊗ f is a local problem. It is equivalent to choosing functions ϕ v ⊗ φ ′ v = θ − ( f ⊗ f ) as just above, by the Shimizu lift θ realising the Jacquet–Langlandscorrespondence (§5.1). For v | p we thus have an explicit choice of such, corresponding to the onemade above. (11) For v ∤ p , we can introduce several restrictions on ( ϕ v , φ v ) as in [60], with theeffect of simplifying many calculations of local heights (§6). Arithmetic theta lifting and kernel identity (§5) . — In [60], the authors introduce an arithmetic-geometric analogue of the Shimizu lift, by means of which they are able to write also the Heegner-points side of their formula as a Petersson product with ϕ of a certain geometric kernel. We canadapt without difficulty their results to reduce our formula to the assertion (12) that d F I ( φ p ∞ ; χ ) − L ( p ) (1 , η ) e Z ( φ ∞ , χ ) is killed by the p -adic Petersson product ℓ ϕ p ,α . Here e Z ( φ ∞ , χ ) is a modular form depending on φ encoding the height pairings of CM points on Shimura curves and their Hecke translates; itgeneralises the classical generating series P m h ι ξ ( P )[ χ ] , T ( m ) ι ξ ( P ) i q m . (10) This is the point which possibly produces poles. (11)
The notation φ ′ refers to the application to φ of a local operator at v | p appearing in the interpolation of thePetersson product. (12) Together with a comparison of local terms at p described below. DANIEL DISEGNI
Decomposition and comparison (§§7-8) . — Both terms in the kernel identity are sums of localterms indexed by the finite places of F . For v ∤ p , we compute both sides and show that thedifference essentially coincides with the one computed in [60]: it is either zero or, at bad places, amodular form orthogonal to all forms in σ . In fact we can show this only for a certain restrictedset of q -expansion coefficients; as the global kernels are p -adic modular forms, this will suffice bya simple approximation argument (Lemma 2.1.2). p -adic Arakelov theory or analytic continuation . — The argument just sketched relies on calcu-lations of arithmetic intersections of CM points; this in general does not suffice, as we need toconsider the contributions of the Hodge classes in the generating series too. It will turn out thatsuch contribution vanishes; two approaches can be followed to show this. The first one, in analogywith [60,61] and already used in a simpler context in [19], is to make use of Besser’s p -adic Arakelovtheory (13) [8] in order to separate such contribution.We will follow an alternative approach (see Proposition 10.2.3), which exploits the generality ofour context and the existence of extra variables in the p -adic world. Once constructed the Heegner–theta element P ± , the anticyclotomic formula of Theorem C.4 is essentially a corollary of TheoremB for all finite order characters χ : we only need to check the compatibility Q v ( f + α,v , f − α,v , χ ) = ζ F,v (2) − · Z v ( χ v ) for all v | p by explicit computation. Conversely, thanks to the multiplicity oneresult, it is also true that Theorem B for any χ is obtained as a corollary of Theorem C.4 byspecialisation. We make use of both of these observations: we first prove Theorem B for allbut finitely many finite order characters χ ; this suffices to deduce Theorem C.4 by an analyticcontinuation argument, which finally yields Theorem B for the remaining characters χ as well.The initially excluded characters are those (such as the trivial character when it is contemplated)for which the contribution of the Hodge classes is not already annihilated by χ -averaging; for allother characters the Arakelov-theoretic arguments just mentioned are then unnecessary. Annihilaton of p -adic heights (§9) . — We are left to deal with the contribution of the places v | p .We can show quite easily that this is zero for the analytic kernel. As in the original work of Perrin-Riou [42], the vanishing of the contribution of the geometric kernel is the heart of the argument.We establish it via an elaboration of a method of Nekovář [40] and Shnidman [52]. The key newingredient in adapting it to our semistable case is a simple integrality criterion for local heights interms of intersections, introduced in §4.3 after a review of the theory of heights. Local toric period . — Finally, in the Appendix we compute the local toric period Q ( θ ( ϕ v ⊗ φ ′ v ) , χ v ) for v | p and compare it to the interpolation factor of the p -adic L -function. Both are highly ramifiedlocal integrals, and they turn out to differ by the multiplicative constant L (1 , η v ) ; this completesthe comparison between the kernel identity and Theorem B. I would like to thank S. Zhang for warm encouragement and A.Burungale, M. Chida, O. Fouquet, M.-L. Hsieh, A. Iovita, S. Kudla, V. Pilloni, E. Urban, Y.Tian, X. Yuan, and W. Zhang for useful conversations or correspondence. I am especially gratefulto S. Shah for his help with the chirality of Haar mesaures and to A. Shnidman for extendeddiscussions on heights. Finally, I would like to thank A. Besser for kindly agreeing to study thecompatibility between p -adic heights and Arakelov theory on curves of bad reduction: his workgave me confidence at a time when I had not found the alternative argument used here.Parts of the present paper were written while the author was a postdoctoral fellow at MSRI,funded under NSF grant 0932078000. We largely follow the notation and conventions of [60, §1.6]. (13)
Recall that an Arakelov theory is an arithmetic intersection theory which allows to pair cycles of any degree,recovering the height pairing for cycles of degree zero. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES L -functions . — In the rest of the paper (and unlike in the Introduction, where we adhere to themore standard convention), all complex L - and zeta functions are complete including the Γ -factorsat the infinite places . (This is to facilitate referring to the results and calculations of [60] wherethis convention is adopted.) Fields and adèles . — The fields E and F will be as fixed in the Introduction unless otherwisenoted. The adèle ring of F will be denoted A F or simply A ; it contains the ring A ∞ of finiteadèles. We let D F and D E be the absolute discriminants of F and E respectively. We also choosean idèle d ∈ A ∞ , × generating the different of F/ Q , and an idèle D ∈ A ∞ , × generating the relativediscriminant of E/F .We use standard notation to restrict adèlic objects (groups, L -functions,. . . ) away from a finiteset of places S , e.g. A S := Q ′ v / ∈ S F v , whereas F S := Q v ∈ S F v . When S is the set of places above p (respectively ∞ ) we use this notation with ‘ S ’ replaced it by ‘ p ’ (respectively ‘ ∞ ’).We denote by F + ∞ ⊂ F ∞ the group of ( x τ ) τ |∞ with x τ > for all τ , and we let A × + := A ∞ , × F + ∞ , F × + := F × ∩ F + ∞ .For a non-archimedean prime v of a number field F , we denote by q F,v the cardinality of theresidue field and by ̟ v a uniformiser. Subgroups of GL . — We consider GL as an algebraic group over F . We denote by P , re-spectively P , the subgroup of GL , respectively SL , consisting of upper-triangular matrices; by A ⊂ P ⊂ GL the diagonal torus; and by N ⊂ P ⊂ GL the unipotent radical of P . We let n ( x ) := ( x ) and w := (cid:18) − (cid:19) . Quadratic torus . — We let T := Res E/F G m ; the embedding T ( A ∞ ) ⊂ B × is fixed. We let Z := G m,F , and view it both as a subgroup of T and as the centre of GL . We let [ T ] := T ( F ) \ T ( A ) /Z ( A ) . Measures . — We choose local and global Haar measures as in [60]. In particular, we have vol( GL ( O F,v )) = | d | v ζ F,v (2) − for all non-archimedean v .We denote by dt the local and global measures on T /Z of [60], which give vol([ T ] , dt ) = 2 L (1 , η ) .The global measure d ◦ t := | D F | / | D E/F | / dt gives vol([ T ] , d ◦ t ) ∈ Q × . We also use the regularised integration R ∗ [ T ] and regularised average − R [ T ] of [60, §1.6.7] to integrate functions on T ( A ) which are not necessarily invariant under Z ( A ) .For functions which factor through T ( A ) /T ( F ∞ ) and are locally constant there, the regularisedintegration reduces to a finite sum and, when using the measure d ◦ t , makes sense for functionstaking p -adic values as well. Multiindices . — If S is a set and r ∈ Z S , p ∈ G S for some group G , we often write p r := Q v ∈ S p r v v .This will typically be applied in the following situation: S = S p is the set of places of F above p , G is the (semi)group of ideals of O F , and p v is the ideal corresponding to v . Functions of p -adic characters . — When Y ? is one of the rigid spaces introduced above and G ( A ) ∈ O ( Y ? ) is a function on Y ? depending on other ‘parameters’ A (e.g. a p -adic L -function),we write G ( A ; χ ) for the evaluation G ( A )( χ ) . DANIEL DISEGNI p -adic modular forms2.1. Modular forms and their q -expansions. — Let K ⊂ GL ( b O F ) be an open compactsubgroup. Recall that a Hilbert automorphic form of level K is a smooth function of moderategrowth ϕ : GL ( F ) \ GL ( A ) /K → C . Let k ∈ Z Hom ( F, R ) . Then an automorphic form is said to be of weight k if it satisfies ϕ ( gr θ ) = ϕ ( g ) ψ ∞ ( k · θ ) for all r θ = (cid:0)(cid:0) cos 2 πθ v sin 2 πθ v − sin 2 πθ v cos 2 πθ v (cid:1)(cid:1) v |∞ ∈ SO ( F ∞ ) . It is said to be holomorphic fo weight k if for all g ∈ GL ( A ∞ ) , the function of z ∞ = ( x v + iy v ) v |∞ ∈ h Hom ( F, R ) , z ∞
7→ | y ∞ | − k/ ∞ ϕ ( g ( y ∞ x ∞ )) is holomorphic. Holomorphic Hilbert automorphic forms will be simply called modular forms.Let ω : F × \ A × → C × be a finite order character. Then ϕ is said to be of character ω if itsatisfies ϕ ( zg ) = ω ( z ) ϕ ( g ) for all z ∈ Z ( A ) ∼ = A × . We denote by M k ( K, C ) the space of modularforms of level K and weight k , and by S k ( K, C ) its subspace of cuspforms. We further denoteby M k ( K, ω, C ) , S k ( K, ω, C ) the subspaces of forms of character ω . We identify a scalar weight k ∈ Z ≥ with the corresponding parallel weight ( k, . . . , k ) ∈ Z Hom ( F, R ) ≥ .For v a finite place of F and N an ideal of O F,v , we define subgroups of GL ( O F,v ) by K ( N ) v = (cid:26)(cid:18) a bc d (cid:19) | c ≡ N (cid:27) ,K ( N ) v = (cid:26)(cid:18) a bc d (cid:19) | c, d − ≡ N (cid:27) ,K ( N ) v = (cid:26)(cid:18) a bc d (cid:19) | c, a − ≡ N (cid:27) ,K ( N ) v = (cid:26)(cid:18) a bc d (cid:19) | c, a − , d − ≡ N (cid:27) ,K ( N ) v = (cid:26)(cid:18) a bc d (cid:19) | b, c, a − , d − ≡ N (cid:27) . If N is an ideal of O F and ∗ ∈ { , , , , ∅} , we define subgroups K ∗ ( N ) of GL ( b O F ) by K ∗ ( N ) = Q v K ∗ ( N ) v . If p is a rational prime and r ∈ Z { v | p }≥ , we further define K ∗ ( p r ) p = Q v K ∗ ( ̟ r v v ) v ⊂ GL ( O F,p ) .Fix a nontrivial character ψ : A /F → C × . Any automorphic form ϕ admits a Fourier–Whittakerexpansion ϕ ( g ) = P a ∈ F W a ( g ) , where W a ( g ) = W ϕ,ψ,a ( g ) satisfies W a ( n ( x ) g ) = ψ ( ax ) W a ( g ) forall x ∈ A . If ϕ is holomoprhic of weight k we can further write W a ( g ) = W a ∞ ( g ∞ ) W a, ∞ ( g ∞ ) with W a, ∞ ( g ) = Q v |∞ W a,v ( g ) , where W a,v = W ( k v ) a,v is the standard holomorphic Whittaker function ofweight k given by (suppressing the subscripts and using the Iwasawa decomposition) W ( k ) a (( z z ) ( y x ) r θ ) = ( | y | k/ ψ ( a ( x + iy )) ψ ( kθ ) R + ( ay ) if a = 0 | y | k/ ψ ( kθ ) R + ( y ) if a = 0 . (2.1.1)(Similarly, we have a description in terms of the standard antiholomorphic Whittaker function W ( − k ) a (( z z ) ( y x ) r θ ) = ( | y | k/ ψ ( a ( x + iy )) ψ ( − kθ ) R + ( − ay ) if a = 0 | y | k/ ψ ( − kθ ) R + ( − y ) if a = 0 (2.1.2)for antiholomorphic forms of weight − k < .) HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES In this case we have an expansion ϕ (cid:18)(cid:18) y x (cid:19)(cid:19) = | y | k/ ∞ X a ∈ F ≥ W ∞ a ( (cid:0) y ∞ (cid:1) ) ψ ∞ ( iay ∞ ) ψ ( ax ) for all y ∈ A × + , x ∈ A ; here F ≥ denotes the set of a ∈ F satisfying τ ( a ) ≥ for all τ : F ֒ → R .For a field L , let the space of formal q -expansions C ∞ ( A ∞ , × , L ) J q F ≥ K ◦ be the set of those formalsums W = P a ∈ F ≥ W a q a with coefficients W a ∈ C ∞ ( A ∞ , × , L ) such that, for some compact subset A W ⊂ A ∞ , we have W a ( y ) = 0 unless ay ∈ A W .Let ϕ be a holomoprhic automorphic form. The expression q ϕ ( y ) := X a ∈ F W ∞ ϕ,a (( y )) q a , y ∈ A ∞ , × (2.1.3)belongs to C ∞ ( A ∞ , × , C ) J q F ≥ K ◦ and it is called the formal q -expansion of ϕ . The space of formal q -expansions is an algebra in the obvious way, compatibly with the algebra structure on automorphicforms. Proposition 2.1.1 ( q -expansion principle) . — Let K ⊂ GL ( b O F ) be an open compact sub-group, and let k ∈ Z Hom ( F, R ) ≥ . The q -expansion map defined by (2.1.3) M k ( K, C ) → C ∞ ( A ∞ , × , C ) J q F ≥ K ◦ ϕ q ϕ is injective. We say that a formal q -expansion is modular if it belongs to the image of the q -expansion map. Proof . — This is (a weak form) of the q -expansion principle of [46, Théorème 6.7 (i)]. In fact ourmodular forms ϕ are identified with tuples ( ϕ c ) c ∈ Cl( F ) + of Hilbert modular forms in the sense of[46]. Then the non-vanishing of q ϕ for ϕ = 0 is obtained by applying the result of loc. cit. to each ϕ c . See [46, Lemme 6.12] for the comparison between various notions of Hilbert modular formsused there.The spaces of formal q -expansions introduced so far will often be convenient for us in terms ofnotation, but they are redundant: if k ∈ Z ≥ , ϕ ∈ M k ( K, C ) , we have W a (( y )) = W (( ay )) forall a ∈ F × , y ∈ A × + . Moreover if K ⊂ K ( N ) , then | y ∞ | − k/ W ( (cid:0) y ∞ (cid:1) ) and | y ∞ | − k/ W ∞ ( (cid:0) y ∞ (cid:1) ) are further invariant under the action of U F ( N ) = { u ∈ b O × F | u ≡ N } by multiplication on y (see [26, Theorem 1.1]). We term reducible of weight k those formal q -expansions satisfying theseconditions for some N .Given a reducible q -expansion W of weight k , we can then define its reduced q -expansion coef-ficients W ♮ ( y ) for y ∈ A ∞ , × /F × + and W ♮a for a ∈ A ∞ , × by W ♮ ( y ) := | y | − k/ W ∞ (( y )) , W ♮a := | a | − k/ W ∞ (( a )) . If A ⊂ C is a subring, we denote by M k ( K, A ) ⊂ M k ( K, C ) , S k ( K, A ) ⊂ S k ( K, C ) the subspacesof forms with reduced q -expansion coefficients in A . If A is any Q -algebra, we let M k ( K, A ) = M k ( K, Q ) ⊗ A , S k ( K, A ) = S k ( K, Q ) ⊗ A . Then it makes sense to talk about the q -expansion ofan element of those spaces. p -adic modular forms . — Let U F ( N p ∞ ) = ∩ r ≥ U F ( N p r ) . We endow the quotient A ∞ , × /F × + U F ( N p ∞ ) with the profinite topology. Let L be a complete Banach ring with norm | · | . We define the spaceof reduced q -expansion coefficients with values in L to be M ′ ( K p ( N ) , L ) := C ( A ∞ , × /F × + U F ( N p ∞ ) , L ) × L A ∞ , × /U F ( Np ∞ ) DANIEL DISEGNI for any nonzero ideal N ⊂ O F prime to p . If K p ⊂ GL ( A p ∞ ) is a compact open subgroup ingeneral, we can define M ′ ( K p , L ) := M ′ ( K p ( N ) , L ) for the largest subgroup K p ( N ) ⊂ K p .We define a norm || · || on M ′ ( K p ( N ) , L ) by || ( W ♮ , ( W ♮a ) a ∈ A ∞ , × /U F ( Np ∞ ) ) || := sup ( y,a ) {| W ♮ ( y ) | , | W ♮a |} . (2.1.4)It induces a norm on the (isomorphic) space of reducible p -adic q -expansions. If ϕ is a modularform, we still denote by q ϕ its reduced q -expansion; in cases where the distinction is significant, theprecise meaning of the expression q ϕ will be clear from its context. We denote by S ′ ( L ) ⊂ M ′ ( L ) the space of reduced q -expansions with vanishing constant coefficients; when there is no risk ofconfusion we shall omit L from the notation.Suppose that L is a field extension of Q p . The space of p -adic modular forms of tame level K p ⊂ GL ( b O pF ) with coefficients in L , denoted by M ( K p , L ) , is defined to be the closure in M ′ ( K p , L ) ,of the subspace generated by the reduced q -expansions of elements of M ( K p K ( p ∞ ) p , L ) = ∪ r ≥ M ( K p K ( p r ) p , L ) . Tame levels and coefficients rings will be omitted from the notationwhen they are understood from context. We denote by S := M ∩ S ′ the space of p -adic modularcuspforms. Approximation . — The q -expansion principle of Proposition 2.1.1 is complemented by the following(obvious) result to provide a p -adic replacement for the approximation argument in [60]. Lemma 2.1.2 (Approximation) . —
Let S be a finite set of finite places of F , not containingany place v above p . Let ϕ be a p -adic modular cuspform all whose reduced q -expansion coefficients W ♮a,ϕ are zero for all a ∈ F × A S ∞ , × . Then ϕ = 0 . Proof . — The form ϕ has some tame level K p ; then its coefficients are invariant under the actionof some compact open U pF ⊂ A ∞ , × on the indices a . Since F × A S ∞ , × U pF = A ∞ , × , the lemmafollows.Let S ′ be the quotient of S ′ by the subspace of reduced q -expansions which are zero at all a ∈ F × A S ∞ , × , and let S be the image of S in S ′ (these notions depend on the set S , which in our useswill be clear from the context). Then the lemma says that in S → S ֒ → S ′ , (2.1.5)the first map is an isomorphism and the composition is an injection. We use the notation S S ( K p ) , S ′ S ( K p ) if we want to specify the set of places S and the tame level K p . Families . — Let Y ? be one of the rigid spaces defined in the Introduction, and K p ⊂ GL ( A p ∞ ) be a compact open subgroup. Definition 2.1.3 . — A Y ? -family of q -expansions of modular forms of tame level K p is a reduced q -expansion ϕ with values in O ( Y ? ) , whose coefficients are algebraic on Y ?l . c . , and such that forevery point χ ∈ Y ?l . c . , ϕ ( χ ) is the reduced q -expansion of a classical modular form ϕ ( χ ) of p level K p K ( p ∞ ) p with coefficients in M ( χ ) . We say that ϕ is bounded if it is bounded for the norm(2.1.4). Twisted modular forms . — It will be convenient to consider the following relaxation of the notionof modular forms.
Definition 2.1.4 . — A twisted
Hilbert automorphic form of weight k ∈ Z Hom ( F, R ) ≥ and level K ⊂ GL ( b O F ) is a smooth function e ϕ : GL ( A ) /K × A × → C satisfying: HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES – for all γ ∈ GL ( F ) , r θ ∈ SO ( F ∞ ) , e ϕ ( γgr θ , u ) = e ϕ ( g, det( γ ) − u ) ψ ∞ ( k · θ ); – e ϕ is of moderate growth in the variable g ∈ GL ( A ) and equal to F + ∞ ( u ∞ ) in the component u ∞ of the variable u = u ∞ u ∞ ; – there exists a a compact open subgroup U F ⊂ A ∞ , × such that for all g , e ϕ ( g, · ) is invariantunder U F ; – for each g ∈ GL ( A ) , there is an open compact subset K g ⊂ A ∞ , × such that e ϕ ( g, · ) issupported in K g F + ∞ .Let ω : F × \ A × → C × be a finite order character. We say that a twisted automorphic form e ϕ hascentral character ω if it satisfies e ϕ ( zg, u ) = ω ( z ) e ϕ ( g, z − u ) for all z ∈ Z ( A ) ∼ = A × . We say that it is holomorphic (of weight k ) or simply a twisted modularform if z ∞
7→ | y ∞ | − k/ ∞ e ϕ ( g ( y ∞ x ∞ ) , u ) is holomorphic in z ∞ = ( x v + iy v ) v |∞ ∈ h Hom ( F, R ) , for all u ∈ A × .We let M tw k ( K, C ) denote the space of twisted modular forms of weight k and M tw k ( K, ω, C ) itssubspace of forms with central character ω . We omit the K from the notation if we do not wishto specify the level.If e ϕ is a twisted modular form, then for each g, u , the function x ϕ ( n ( x ) g, u ) descends to F \ A and therefore it admits a Fourier–Whittaker expansion in the usual way. To the restrictionof e ϕ to GL ( A ) × F × we then attach a twisted formal q -expansion X a ∈ F | y | k/ ∞ W ∞ a (( y ) , u ) q a ∈ C ∞ ( A ∞ , × × F × , C ) J q F ≥ K ◦ such that e ϕ (cid:18)(cid:18) y x (cid:19) , u (cid:19) = | y | k/ ∞ X a ∈ F ≥ W ∞ a ( (cid:0) y ∞ (cid:1) , u ) ψ ∞ ( iay ∞ ) ψ ( ax ) for all y ∈ A × + , x ∈ A , u ∈ F × . Here the space C ∞ ( A ∞ , × × F × , C ) J q F ≥ K ◦ consists of q -expansions W whose coefficients W a ( y, u ) vanish for ay outside of some compact open subset A W ⊂ A ∞ .Let e ϕ be a twisted modular form, let U F ⊂ A ∞ , × be a compact open subgroup satisfying thecondition of the previous definition, and let µ U F = F × ∩ U F . Then the sum ϕ ( g ) := X u ∈ µ UF \ F × e ϕ ( g, u ) is finite for each g (if K g ⊂ A ∞ , × is a compact subset such that K g F ×∞ contains the support of e ϕ ( g, · ) , the sum is supported on µ U F \ ( F × ∩ K g ) , which is commensurable with the finite group µ U F \ O × F ). It defines a modular form in the usual sense, with formal q -expansion q ϕ ( y ) = X u ∈ µ UF \ F × q e ϕ ( y, u ) . One can, similarly to the above, define a norm on the space of twisted formal q -expansion coeffi-cients of a fixed parallel weight k with values in a Banach ring L , namely || W || := sup ( a,y ∞ ,u ) | y ∞ | − k/ | W ∞ a ( y, u ) | .The p -adic completion M tw k ( K p , L ) of the subspace of q -expansions of twisted modular forms (ofsome tame level K p ) is called the space of p -adic twisted modular forms (of tame level K p ). Finally,there is a notion of Y ? -family of q -expansions of twisted modular forms. DANIEL DISEGNI U v . — Let H = C ∞ c ( GL ( A ∞ ) , Q ) be the rational Hecke algebra of smooth compactly supported functions with the convolution opera-tion (denoted by ∗ ), and for any finite set of non-archimedean places S let H S = C ∞ c ( GL ( A S ∞ ) , Q ) , H S = C ∞ c ( GL ( F S ) , Q ) .The group GL ( A ∞ ) has a natural left action on automoprhic forms by right multiplication.This action is extended to elements f ∈ H ⊗ C by T ( f ) ϕ ( g ) = Z GL ( A ∞ ) f ( h ) ϕ ( gh ) dh, where dh = Q dh v with dh v the Haar measure on GL ( F v ) assigning volume to GL ( O F,v ) . If K ⊂ GL ( A ∞ ) is a compact open subgroup, we define e K = T (vol( K ) − K ) ∈ H . It acts asa projector on K -invariant forms. If g ∈ GL ( A ∞ ) and K , K ′ ⊂ GL ( b O F ) are open compactsubgroups, we define the operator [ KgK ′ ] := T ( KgK ′ ) .By the strong multiplicity one theorem, for each level K , each M -rational automorphic rep-resentation σ which is discrete series of weight at all infinite places, and each finite set ofnon-archimedean places S such that K is maximal away from S , there are spherical (that is, K (1) S -biinvariant) elements T ( σ ) ∈ H S ⊗ M whose action on M ( K, M ) is given by the idempo-tent projection e σ onto σ K ⊂ M ( K, M ) .On the space M ( K p , L ) of p -adic modular forms, with K p ⊃ K ( N ) p , there is a continuous actionof Z ( N p ∞ ) := A ∞ , × /F × U F ( N p ∞ ) , extending the central action z.ϕ ( g ) = ϕ ( gz ) on modularforms. For a continuous character ω : Z ( N p ∞ ) → L × , we denote by M ( K p , ω, L ) the set of p -adic modular forms ϕ satisfying z.ϕ = ω ( z ) ϕ , and by S ( K p , ω, L ) its subspace of cuspidalforms. If ω is the restriction of a finite order character of Z ( A ∞ ) /U F ( N p ∞ ) , then we have M ( K p K ( p ∞ ) p , ω, L ) ⊂ M ( K p , ω, L ) .The action of H Sp = C ∞ c ( GL ( A Sp ∞ ) , Q ) extends continuously to the space S ( K p , ω, L ) if K p is maximal away from S ; explicitly, if ϕ is the q -expansion with reduced coefficients W ♮a,ϕ and h ( x ) = K (1) Np ( ̟ v ) K (1) Np , we have W ♮a,T ( h ) ϕ = W ♮a̟ v ,ϕ + ω − ( ̟ v ) W ♮a/̟ v ,ϕ . (2.2.1)Moreover if S ′ is another set of finite places not containing those above p and S ′′ = S ∪ S ′ , theaction of H S ′′ p extends in the same way to the space S ′ = S ′ S ′ ( K p ) defined after Lemma 2.1.2. Operators U v . — Let v be a finite place of F , ̟ v ∈ F v a uniformiser, K v ⊂ GL ( b O vF ) a compactopen subgroup. For each r ≥ , we define Hecke operators U ∗ v,r = [ K v K ( ̟ rv ) v ( ̟ v ) K v K ( ̟ rv ) v ] , U v, ∗ ,r = [ K v K ( ̟ rv ) v (cid:16) ̟ − v (cid:17) K v K ( ̟ rv ) v ] . They depend on the choice of uniformisers ̟ v , although a sufficiently high (depending on r ) integerpower of them does not. They are compatible with changing r in the sense that U v, ∗ ,r e K ( ̟ r ′ v ) v =U v, ∗ r ′ for r ′ ≤ r ; we will hence omit the r from the notation. If ϕ ∈ S ( K p K ( p r ) p , ω ) has re-duced q -expansion coefficients W ♮ϕ,a for a ∈ A ∞ , × , then U v, ∗ ϕ has reduced q -expansion coefficients W ♮ U v, ∗ ϕ,a = ω − ( ̟ v ) W ♮ϕ,aω v . By this formula we can extend U v, ∗ to a continuous operator on p -adic reduced q -expansions, and in particular on p -adic modular forms. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES Atkin-Lehner operators . — Let v be a finite place and fix the same uniformiser ̟ v as in theprevious paragraph. Then we define elements w r,v := (cid:18) − ̟ rv (cid:19) ∈ GL ( F v ) ⊂ GL ( A ) for r ≥ , and denote by the same names the operators they induce on automorphic forms by rightmultiplication. We have w − r,v K ( ̟ s ) v w r,v = K ( ̟ sv ) v .If r = ( r v ) v | p , we define w r = ( w r v ,v ) v | p ∈ GL ( F p ) = Q v | p GL ( F v ) , and similarly w − r . Let F v be a non-archimedean local field,and recall the space Ψ v of abstract additive characters of level of F v defined in §1.2. Let ψ univ ,v : F v → O (Ψ v ) × be the tautological character, which we identify with an action of theunipotent subgroup N = N ( F v ) ∼ = F v ⊂ GL ( F v ) on the sheaf O Ψ v . Let σ v be an infinite-dimensional representation of GL ( F v ) on a vector space over a number field M . A Whittakermodel over M ⊗ O Ψ v for σ v ⊗ Q O Ψ v is a non-trivial GL ( F v ) -equivariant map σ v ⊗ O Ψ v → M ⊗ Ind GL ( F v ) N ψ univ ,v of free sheaves over M ⊗ O Ψ v . We will often identify this map with itsimage.Let P ⊂ GL ( F v ) be the mirabolic group of matrices ( a b ) . A Kirillov model over M ⊗ O Ψ v for σ v ⊗ Q O Ψ v is a non-trivial P -equivariant map σ v ⊗ O Ψ v → M ⊗ Ind P N ψ univ ,v . We will oftenidentify this map with its image and the image with a subsheaf of C ∞ ( F × v , M ) ⊗ O Ψ v by restrictingfunctions from P to { ( a ) | a ∈ F × v } ∼ = F × v . Lemma 2.3.1 . —
Let σ v be an infinite-dimensional representation of GL ( F v ) on a rational vec-tor space, M = End ( σ v ) . Then σ v ⊗ Q O Ψ v admits a Whittaker model W ( σ v , ψ univ ,v ) (respectively,a Kirillov model K ( σ v , ψ univ ,v ) ) over M ⊗ O Ψ v , unique up to ( M ⊗ O Ψ v ) × , whose specialisationat every closed point ψ v ∈ Ψ v is the unique Whittaker model W ( σ v , ψ v ) (respectively, the uniqueKirillov model K ( σ v , ψ v ) ) of σ v ⊗ Q ( ψ v ) .If we view W ( σ v , ψ univ ,v ) (respectively K ( σ v , ψ univ ,v ) ) as a subsheaf of C ∞ ( GL ( F v ) , M ) ⊗ O Ψ v (respectively as a subsheaf of C ∞ ( F × v , M ) ⊗ O Ψ v ), then the restriction map W f , f ( y ) := W (( y )) induces an isomorphism W ( σ v , ψ univ ,v ) → K ( σ v , ψ univ ,v ) .We call W ( σ v , ψ univ ,v ) (respectively K ( σ v , ψ univ ,v ) ) the universal Whittaker model (respectively, the universal Kirillov model ) for σ v . The universal Kirillov model admits a natural M -structure,that is an M -vector space (14) K ( σ v , ψ univ ,v ) M ⊂ C ∞ ( F × v , M ) such that K ( σ v , ψ univ ,v ) M ⊗ O Ψ v = K ( σ v , ψ univ ,v ) .Proof . — The proof of existence and uniqueness of Whittaker models given e.g. in [9, §4.4] carriesover to our context after replacing C by M ⊗ O Ψ v and the fixed C × -valued character ψ v of loc. cit. with ψ univ ,v . The analogous result for Kirillov models, together with the isomorphism W ( σ v , ψ univ ,v ) → K ( σ v , ψ univ ,v ) , follows formally from Frobenius reciprocity as in [11, Corollary36.2]. We prove the assertion on the M -structure for K ( σ v , ψ univ ,v ) , after dropping subscripts v .As in the classical case, the space of Schwartz functions S ( F × , M ) ⊗ O Ψ is an irreducible P -representation (see [11, Corollary 8.2]), hence contained in K := K ( σ ι , ψ univ ) ⊂ C ∞ ( F × , M ) ⊗ O Ψ . Moreover K := K / S ( F × , M ) ⊗ O Ψ is a free sheaf over M ⊗ O Ψ of rank d ≤ dependingon the type of σ (as can be checked on the points of Ψ by the classical theory). Since the space S ( F × , M ) ⊗ O Ψ has the obvious M -structure S ( F × , M ) , it suffices to describe d generators for K represented by functions in C ∞ ( F × , M ) . (14) Which is not stable under the action of GL ( F v ) . DANIEL DISEGNI If σ is supercuspidal then d = 0 and there is nothing to prove. If σ = St( µ | · | − ) is special with M × -valued central character µ |·| − , then d = 1 and a generator for K is f µ ( y ) := µ ( y ) O F −{ } ( y ) .If σ is an irreducible principal series Ind( µ , µ | · | − ) (plain un-normalised induction) with M × -valued characters µ , µ , then d = 2 ; if µ = µ , a pair of generators for K is { f µ , f µ } . If µ = µ = µ , a pair of generators is { f µ , f ′ µ } with f ′ µ ( y ) := v ( y ) µ ( y ) O F −{ } ( y ) . We will often slightly abusively identify Whittaker and Kirillov models by W f , f ( y ) = W (( y )) .If σ ∞ is an M -rational automorphic representation of weight , then after choosing any em-bedding ι : M ֒ → C and any nontrivial character ψ : A /F → C × , the q -expansion coefficients ofany ϕ ∈ σ ∞ can be identified with the product of the local Kirillov-restrictions f v of the Whit-taker function W = W v of ϕ ι (when W is indeed factorisable). Equivalently, the f v belong to the M -rational subspaces and are therefore independent of the choice of additive character. Lemma 2.3.2 . —
In the situation of the previous lemma, there is a pairing ( , ) v : K ( σ v , ψ univ ,v ) ⊗ M K ( σ ∨ v , ψ univ ,v ) → M ⊗ O Ψ v such that for any f , f in the M -rational subspaces K ( σ v , ψ univ ,v ) M , respectively K ( σ ∨ v , ψ v ) M ,the paring ( f , f ) v ∈ M , and that for any ι : M ֒ → C , we have ι ( f , f ) v = ζ F,v (2) L (1 , σ ιv × σ ι ∨ v ) Z F × v ιf ( y ) ιf ( y ) d × y | d | / v . (2.3.1)The right-hand side is understood in the sense of analytic continuation to s = 0 for the functionof s defined, for ℜ ( s ) sufficiently large, by the normalised convergent integral ζ F,v (2) L (1 + s, σ ιv × σ ι ∨ v ) Z F × ιf ( y ) ιf ( y ) | y | s d × y | d | / v . The normalisation is such that the pairing equals when σ v is an unramified principal seriesand the f i are normalised new vectors. Proof . — We use the notation of the proof of Lemma 2.3.1, dropping all subscripts v . We simplyneed to show that the given expression belongs to ιM if f , f belong to the M -rational subspaceof K and that any pole of the integral I s ( f , f ) := R F × ιf ( y ) ιf ( y ) | y | s d × y | d | / is cancelled by a poleof L (1 + s, σ ι × σ ι ∨ ) . If either of f i ∈ S ( F × , M ) , the integral is just a finite sum of elements in ιM . Then we only need to compute the integral when f , f are among the M -rational generatorsof K , which is a standard calculation.In our application there will be no poles by the Weil conjectures, so we limit ourselves to provingthe statement in the case where σ = Ind( µ , µ | · | − ) is a principal series with µ = µ . (Theother cases are similar, cf. also the proof of Proposition 3.6.1.) Then σ ∨ = Ind( µ ′ , µ ′ | · | − ) with µ ′ = µ − | · | , µ ′ = µ − | · | , and (dropping also the ι from the notation) L (1 + s, σ × σ ∨ ) =(1 − q − − sF ) − (1 − µ µ ′ ( v ) q − sF ) − (1 − µ ′ µ ( v ) q − sF ) − , where µ ( v ) := µ ( ̟ v ) if µ is an unramifiedcharacter and µ ( v ) := 0 otherwise.Assume that f = f µ (the case f = f µ is similar). If f = f µ ′ then I s ( f , f ) = (1 − q − − sF ) − has no pole at s = 0 . If f = f µ ′ , then I s ( f , f ) = (1 − µ µ ′ ( v ) q − sF ) − , whose inverse is a factorof L (1 + s, σ v × σ ∨ v ) − in M [ q − sF ] . p -crtical forms and the p -adic Petersson product. — As in [19], we introduce thefollowing notion.
Definition 2.4.1 . — Let W = (0 , ( W a )) be a reduced q -expansion without constant term, withvalues in a p -adic field L , and let v | p . We say that W is v -critical if there is c ∈ Z such that, for HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES each a ∈ F × A ∞ , W a̟ sv ∈ q s − cF,v O L . We say that W is p -critical if it is a sum of v -critical q -expansions for v | p .For each v | p , we define ordinary projectors e v and e on M ( K p , ω, L ) by e v ( ϕ ′ ) = lim n →∞ U n ! v, ∗ ϕ ′ , e := Y v e v . They are independent of the choice of uniformisers. The image of e v is contained in M ( K p K ( p ∞ ) , ω, L ) . It is clear that v -critical forms belong to the kernel of e v . p -adic Petersson product . — Let M be a number field, and let σ ∞ be an M -rational cuspi-dal automorphic representation of GL of weight as in Definition 1.2.1, with central character ω : F × \ A → M × .Following Hida, we will define a p -adic analogue of the Petersson inner product with a form ϕ in σ ∞ when σ p is p -ordinary for a prime p | p of M . First we define an algebraic version of the Peterssonproduct, which requires no ordinariness assumption. If ι : M ֒ → C , let ϕ ι := ιϕ ⊗ ϕ ∞ ∈ σ ι be theautomorphic form whose Whittaker function at infinity is anti-holomorphic of smallest K ∞ -type. Lemma 2.4.2 . —
There is a unique pairing ( , ) σ ∞ : σ ∞ ⊗ M M ( ω − , M ) → M such that for any ϕ ∈ σ ∞ , ϕ ∈ M ( ω − , M ) , and any ι : M ֒ → C , we have ( ϕ , ϕ ) σ = | D F | / ζ F (2) L (1 , σ ι , ad) ( ϕ ι , ιϕ ) , where ( ϕ ′ , ϕ ′ ) := Z GL ( F ) Z ( A ) \ GL ( A ) · ϕ ′ ( g ) ϕ ′ ( g ) dg is the usual Petersson product on complex automorphic forms with respect to the Tamagawa masure dg .Proof . — Note first that if such a pairing exists, it annihilates forms on the right-hand side whichare orthogonal (under the complex Petersson product in any embedding) to forms in σ . Then wejust need to use a well-known formula for the adjoint L -value in terms of Petersson product; wequote it in the version given in [53, p. 55]: for an antiholomorphic form ϕ ′ in the space of σ ι anda holomorphic form ϕ ′ in the space of σ ∨ ,ι , both rational over ιM , with factorisable Whittakerfunctions W ιi , we have | D F | / ζ F (2)( ϕ ′ , ϕ ′ )2 L (1 , σ ι , ad) = Y v ( W ι ,v , W ι ,v ) v (2.4.1)where for all v the local pairings are given by the right-hand side of (2.3.1) and do not depend onthe choice of additive characters. Each local factor in the product is rational over ιM and almostall of them are equal to . Remark 2.4.3 . — If ϕ ∈ M ( M ) does not have central character ω − , we can still define ( ϕ , ϕ ) σ ∞ := ( ϕ , ϕ ,ω − ) σ ∞ where ϕ ,ω − ( g ) := − Z Z ( F ) \ Z ( A ) ϕ ( zg ) ω ( z ) dz. Now fix a prime p | p of M and a finite extension L of M p , and assume that for all v | p , σ v ⊗ L is nearly p -ordinary with unit character α v : F × v → O × L in the sense of Definition 1.2.2. Fix a DANIEL DISEGNI
Whittaker functional W p = Q v W v at p and let ϕ ∈ σ ∞ ⊗ M M ( α ) be a form in the space of σ ∞ whose image under W v is the function (viewed in the M ( α ) -rational part of any Kirillov model) W v ( y ) = O F −{ } ( y ) | y | α v ( y ) . (2.4.2)Note that W v , viewed in a Kirillov model associated to an additive character of level , satisfies U ∗ v W v = α v ( ̟ v ) W v .In the next Proposition, we use the notation α ( ̟ ) r := Q v | p α v ( ̟ v ) r v . Proposition 2.4.4 . —
There exists a unique bounded linear functional ℓ ϕ p ,α : M ( K p , ω − , L ) → L satisfying the following:1. Let r = ( r v ) v | p ∈ Z { v | p }≥ . The restriction of ℓ ϕ p ,α to M ( K p K ( p r ) p , M ( α )) is given by (2.4.3) ℓ ϕ p ,α ( ϕ ′ ) = α ( ̟ ) − r ( w r ϕ, ϕ ′ ) σ ∞ = α ( ̟ ) − r ( ϕ, w − r ϕ ′ ) σ ∞ ∈ M ( α ) for any choice of uniformisers ̟ v in the definitions of U v, ∗ , U ∗ v , w r .2. We have ℓ ϕ p ,α (U v, ∗ ϕ ′ ) = α v ( ̟ v ) ℓ ϕ p ,α ( ϕ ′ ) for all v | p and all ϕ ′ .3. ℓ ϕ p ,α vanishes on p -critical forms.4. Let T ( σ ∨ ) ∈ H SM (where S is any sufficiently large set of finite places containing thosedividing p ) be any element whose image T ( σ ) ι ∈ H SM ⊗ M,ι C acts on S ( K p K ( p r ) p , C ) asthe idempotent projector onto ( σ ∨ ,ι ) K p K ( p r ) p for any ι : M ֒ → C and r ≥ . Let T ι p ( σ ∨ ) bethe image of T ( σ ∨ ) in H SM ⊗ M,ι p L . Then ℓ ϕ p ,α ◦ T ι p ( σ ∨ ) = ℓ ϕ p ,α . Proof . — By property 2, for each v we must have(2.4.4) ℓ ϕ p ,α ( e v ϕ ′ ) = lim n →∞ ℓ ϕ p ,α (U n ! v, ∗ ϕ ′ ) = lim n →∞ α v ( ̟ v ) n ! ℓ ϕ p ,α ( ϕ ′ ) = ℓ ϕ p ,α ( ϕ ′ ) as α v ( ̟ v ) is a p -adic unit; note that this expression does not depend on the choice of uniformisers.It follows that ℓ ϕ p ,α must factor through the ordinary projection e : M ( K p , ω − , L ) → M ( K p K ( p ∞ ) p , ω − , L ) , which implies property 3. On the image of e , ℓ ϕ p ,α must be defined defined by (2.4.3), which makesuniqueness and property 4 clear.It remains to show the existence (that is, that (2.4.3) is compatible with changing r ) and thatthe first equality in (2.4.3) holds for all r for the functional ℓ ϕ p ,α just defined (the second one istrivial). For the latter, we have(2.4.5) ( w r ϕ, U v, ∗ ϕ ′ ) = ( w r ϕ, K ( ̟ r v v ) v (cid:16) ̟ − v (cid:17) ϕ ′ ) = ( (cid:0) ̟ v (cid:1) K ( ̟ r v v ) v w r ϕ, ϕ ′ )= ( w r K ( ̟ r v v ) v ( ̟ v ) ϕ, ϕ ′ ) = ( w r U ∗ v ϕ, ϕ ′ ) = α v ( ̟ v )( w r ϕ, ϕ ′ ) . The compatibility with change of r can be seen by a similar calculation.We still use the notation ℓ ϕ p ,α for the linear form deduced from ℓ ϕ p ,α by extending scalars to some L -algebra. The analogous remark will apply to ( , ) σ ∞ .
3. The p -adic L -function3.1. Weil representation. — We start by recalling from [56, 60] the definition of the Weilrepresentation for groups of similitudes. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES Local case . — Let V = ( V, q ) be a quadratic space of even dimension over a local field F ofcharacteristic not 2. Fix a nontrivial additive character ψ of F . For simplicity we assume V haseven dimension. For u ∈ F × , we denote by V u the quadratic space ( V, uq ) . We let GL ( F ) × GO ( V ) act on the usual space of Schwartz functions (15) S ( V × F × ) as follows (here ν : GO ( V ) → G m denotes the similitude character): – r ( h ) φ ( x, u ) = φ ( h − x, ν ( h ) u ) for h ∈ GO ( V ) ; – r ( n ( b )) φ ( x, u ) = ψ ( buq ( x )) φ ( x, u ) for n ( b ) ∈ N ( F ) ⊂ GL ( F ) ; – r (cid:18)(cid:18) a d (cid:19)(cid:19) φ ( x, u ) = χ V u ( a ) | ad | dim V φ ( at, d − a − u ) ; – r ( w ) φ ( x, u ) = γ ( V u ) ˆ φ ( x, u ) for w = (cid:18) − (cid:19) . Here χ V = χ ( V,q ) is the quadratic character attached to V , γ ( V, q ) is a fourth root of unity, and ˆ φ denotes Fourier transform in the first variable with respect to the self-dual measure for thecharacter ψ u ( x ) = ψ ( ux ) . We will need to note the following facts (see e.g. [32]): χ V is trivial if V is a quaternion algebra over F or V = F ⊕ F , and χ V = η if V is a separable quadratic extension E of F with associated character η ; and γ ( V ) = +1 if V is the space of × matrices or V = F ⊕ F , γ ( V ) = − if V is a non-split quaternion algebra.We state here a lemma which will be useful later. Lemma 3.1.1 . —
Let F be a non-archimedean local field and φ ∈ S ( V × F × ) a Schwartz functionwith support contained in { ( x, u ) ∈ V × F × : uq ( x ) ∈ O F } . Suppose that the character ψ used to construct the Weil representation has level . Then φ isinvariant under K ( ̟ r ) ⊂ GL ( O F ) for sufficiently large r . If moreover φ ( x, u ) only depends on x and on the valuation v ( u ) , then φ is invariant under K ( ̟ r ) .Proof . — By continuity of the Weil representation, for the first assertion it suffices to show theinvariance under N ( O F ) . This follows from the observation that under our assumption, in theformula r ( n ( b )) φ ( x, u ) = ψ ( ubq ( x )) φ ( x, u ) , the multiplier ψ ( ubq ( x )) = 1 whenever ( x, u ) is in the support of φ . The second assertion is thenequivalent to the invariance of φ under the subgroup (cid:16) O × F (cid:17) ⊂ GL ( O F ) , which is clear. Fock model and reduced Fock model . — Assume that F = R and V is positive definite. Then wewill prefer to consider a modified version of the previous setting. Let the Fock model S ( V × R × ) be the space of functions spanned by those of the form H ( u ) P ( x ) e − π | u | q ( x ) , where H is a compactly supported smooth function on R × and P is a complex polynomial functionon V . This space is not stable under the action of GL ( R ) , but it is so under the restriction ofthe induced ( gl , R , O ( R )) -action on the usual Schwartz space (see [60, §2.1.2]).We will also need to consider the reduced Fock space S ( V × R × ) spanned by functions of theform φ ( x, u ) = ( P ( uq ( x )) + sgn( u ) P ( uq ( x ))) e ∓ π | u | q ( x )) where P , P are polynomial functions. It contains the standard Schwartz function φ ( x, u ) = R + ( u ) e − π | u | ( q ( x )) , (15) The notation is only provisional for the archimedean places, see below. DANIEL DISEGNI which for x = 0 satisfies r ( g ) φ ( x, u ) = W ( d ) uq ( x ) ( g ) (3.1.1)if V has dimension d and W ( d ) is the standard holomorphic Whittaker function (2.1.1) (see[60, §4.1.1]).There is a surjective quotient map(3.1.2) S ( V × R × ) → S ( V × R × )Φ φ = Φ = Z R × − Z O ( V ) r ( ch )Φ dh dc. For the sake of uniformity, when F is non-archimedean we set S ( V × F × ) := S ( V × F × ) . Global case . — Let ( V , q ) be an even-dimensional quadratic space over the adèles A = A F of atotally real number field F , and suppose that V ∞ is positive definite; we say that V is coherent ifit has a model over F and incoherent otherwise. Given an b O F -lattice V ⊂ V , we define the space S ( V × A × ) as the restricted tensor product of the corresponding local spaces, with respect to thespherical elements φ v ( x, u ) = V v ( x ) ̟ nvv ( u ) , if ψ v has level n v . We call such φ v the standard Schwartz function at a non-archimedan place v .We define similarly the reduced space S ( V × A × ) , which admits a quotient map S ( V × A × ) → S ( V × A × ) (3.1.3)defined by the product of the maps (3.1.2) at the infinite places and of the identity at the finiteplaces. The Weil representation of GL ( A ∞ ) × GO ( V ∞ ) × ( gl ,F ∞ , O ( V ∞ )) is the restrictedtensor product of the local representations. Let V be a two-dimensional quadratic space over A F , totally definiteat the archimedean places. Consider the Eisenstein series E r ( g, u, φ , χ F ) = X γ ∈ P ( F ) \ SL ( F ) δ χ F ,r ( γgw r ) r ( γg ) φ (0 , u ) where δ χ F ,r ( g ) = ( χ F ( d ) − if g = (cid:0) a bd (cid:1) k with k ∈ K ( p r )0 if g / ∈ P K ( p r ) . and φ ∈ S ( V × A × ) . (The defining sum is in fact not absolutely convergent, so it must beinterpreted in the sense of analytic continuation at s = 0 from the series obtained by replacing δ χ F,r with δ χ F,r δ s , where δ s ( g ) = | a/d | s if g = (cid:0) a bd (cid:1) k , k ∈ K (1) .) It belongs to the space M tw1 ( ηχ − F , C ) of twisted modular form of parallel weight and central character ηχ − F .After a suitable modification, we study its Fourier–Whittaker expansion and show that it inter-polates to a Y F -family of q -expansions of twisted modular forms. Proposition 3.2.1 . —
We have L ( p ) (1 , ηχ F ) E r ( (cid:0) y x (cid:1) , u, φ , χ F ) = X a ∈ F W a,r ( (cid:0) y (cid:1) , u, φ , χ F ) ψ ( ax ) where W a,r ( g, u, φ , χ F ) = Y v W a,r,v ( g, u, φ ,v , χ F,v ) HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES with, for each v and a ∈ F v , W a,r,v ( g, u, φ ,v , χ F,v ) = L ( p ) (1 , η v χ F,v ) × Z F v δ χ F ,v,r ( wn ( b ) gw r ) r ( wn ( b ) g ) φ ,v (0 , u ) ψ v ( − ab ) db. Here L ( p ) ( s, ξ v ) := L ( s, ξ v ) if v ∤ p and L ( p ) ( s, ξ v ) := 1 if v | p , and we convene that r v = 0 if v ∤ p .Proof . — The standard expansion of Eisenstein series reads E r ( gw r , u, φ , χ F ) = δ χ F ,r ( gw r ) r ( g ) φ (0 , u ) + X a ∈ F W ∗ a,r ( g, u, φ , χ F ) ψ ( ax ) where W ∗ a,r = L ( p ) (1 , ηχ F ) − W a,r ; but it is easy to check that δ χ F,r ( (cid:0) y x (cid:1) w r ) = 0 .We choose convenient normalisations for the local Whittaker functions: let γ u,v = γ ( V ,v , uq ) be the Weil index, and for a ∈ F × v set W ◦ a,r,v ( g, u, φ ,v , χ F,v ) := γ − u,v W a,r,v ( g, u, φ ,v , χ F,v ) . For the constant term, set W ◦ ,r,v ( g, u, φ ,v , χ F,v ) := γ − u,v L ( p ) (0 , η v χ F,v ) W ,r,v ( g, u, φ ,v , χ F,v ) . Then for the global Whittaker functions we have W a,r ( g, u, φ , χ F ) = − ε ( V ) Y v W ◦ a,r,v ( g, u, φ ,v , χ F,v ) (3.2.1)if a ∈ F × , where ε ( V ) = Q v γ u,v equals − if V is coherent or +1 if V is incoherent; and W ,r ( g, u, φ , χ F ) = − ε ( V ) L ( p ) (0 , ηχ F ) Y v W ◦ ,r,v ( g, u, φ ,v , χ F,v ) . (3.2.2)We sometimes drop φ from the notation in what follows. Lemma 3.2.2 . —
For each finite place v and y ∈ F × v , x ∈ F v , u ∈ F × v , we have W a,v ( (cid:0) y x (cid:1) , u ) = ψ v ( ax ) χ F ( y ) − | y | / W ay,v (1 , y − u ) . The proof is an easy calculation.
Proposition 3.2.3 . —
The local Whittaker functions satisfy the following.1. If v ∤ p ∞ , then W ◦ a,v,r = W ◦ a,v does not depend on r , and for all a ∈ F v W ◦ a,v (1 , u, χ F ) = | d v | / L (1 , η v χ F,v )(1 − χ F,v ( ̟ v )) ∞ X n =0 χ F,v ( ̟ v ) n q nF,v Z D n ( a ) φ ,v ( x , u ) d u x , where d u x is the self-dual measure on ( V ,v , uq ) and D n ( a ) = { x ∈ V ,v | uq ( x ) ∈ a + p nv d − v } . (When the sum is infinite, it is to be understood in the sense of analytic continuation fromcharacters χ F | · | s with s > , cf. the proof of Lemma 3.3.1 below.)2. If v | p , and φ ,v is the standard Schwartz function, then W ◦ a,r,v (1 , u, χ F ) = ( | d v | / | D v | / χ F,v ( − if v ( a ) ≥ − v ( d v ) and v ( u ) = − v ( d v )0 otherwise. DANIEL DISEGNI
3. If v |∞ and φ ,v is the standard Schwartz function,then W a,v (1 , u ) = e − πa if ua > if a = 00 if ua < . Proof . — Part 1 is proved similarly to [60, Proposition 6.10 (1)], whose Whittaker function W ◦ a,v ( s, , u ) equals our L (1 , η v | · | sv ) − W ◦ a,v (1 , u, | · | sv ) . The proof of Part 2 is similar to thatof [19, Proposition 3.2.1, places | M/δ ]. Part 3 is also well-known, see e.g. [60, Proposition 2.11]whose normalisation differs from ours by a factor of γ v L (1 , η v ) − = πi . Lemma 3.2.4 . —
Let a ∈ F . For all finite places v , | d | − / v | D v | − / W ◦ a,v (1 , u, χ F ) ∈ Q [ χ F , φ v ] ,and for almost all v we have | d | − / v | D v | − / W ◦ a,v (1 , u, χ F ) = ( if v ( a ) ≥ − v ( d v ) and v ( u ) = − v ( d v )0 otherwise.Proof . — This follows from Proposition 3.2.3.1 by an explicit computation which is neither difficultnor unpleasant: we leave it to the reader. Recall from §1.2 the profinite groups Γ and Γ F and the associatedrigid spaces Y ′ , Y , Y F (only the latter is relevant for this subsection). For each finite place v ∤ p of F , there are local versions Y ′ v , Y v , Y F,v (3.3.1)which are schemes over M representing the corresponding spaces of G m,M -valued homomorphismswith domain E × v / ( V p ∩ E × v ) (for Y ′ v , Y v , where V p ⊂ E × A p ∞ is the subgroup fixed in the intro-duction) or F × v (for Y F,v ). (16) Letting N ′ denote the restricted tensor product with respect to theconstant function , and the symbol Y ? stand for any of the symbols Y ′ , Y , Y F , we let O Y ? ( Y ? ) f ⊂ O Y ? ( Y ? ) b denote the image of N ′ v ∤ p O ( Y ? v ) ⊗ M L → O Y ? ( Y ? ) . Lemma 3.3.1 . —
For each a ∈ F , y ∈ A ∞ , × and rational Schwartz function φ p ∞ , there are:1. for each v ∤ p ∞ :(a) a Schwartz function φ ,v ( · ) ∈ S ( V ,v , O ( Y F,v )) such that φ ,v ( ) = φ ,v and φ ,v ( · ) isidentically equal to φ ,v if φ ,v is standard;(b) a function W ◦ a,v ( y v , u, φ ,v ) ∈ O Y F,v ( Y F,v ) satisfying W ◦ a,v ( y v , u, φ ,v ; χ F ) = | d v | − / | D v | − / W ◦ a,r,v (( y v ) , u, φ ,v ( χ F ) , χ F ) for all χ F,v ∈ Y F,v ( C ) ;2. a global function W a ( y, u, φ p ∞ ) ∈ O Y F ( Y F ) b , which is algebraic on Y l . c .F and satisfies W a ( y, u, φ p ∞ ; χ F ) = | D F | / | D E | / W ∞ a,r (( y ) , u, φ ( χ F ) , χ F ) (16) Concretely they are closed subschemes of split tori over M , cf. the proof of Proposition 3.6.1. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES for each χ F ∈ Y l . c .F ( C ) ; here φ ( χ F ) = Q v ∤ p ∞ φ ,v ( χ F,v ) φ p ∞ with φ v = φ v the standardSchwartz function for each v | p ∞ . The function | y | − / W a ( y, u, φ p ∞ ) is bounded solely interms of max | φ p ∞ | , and if a = 0 , then W a ∈ O ( Y F ) f .Proof . — If a = 0 , by Lemma 3.2.4 and Proposition 3.2.3.2, we can deduce the existence of theglobal function in part 2 from the local result of part 1. If a = 0 , then by (3.2.2) the same istrue thanks to the well-known existence [18] of a bounded analytic function on Y F interpolating χ F L ( p ) (0 , ηχ F ) .It thus suffices to prove part 1, and moreover we may restrict to y = 1 in view of Lemma3.2.2. We can uniquely write φ ,v = cφ ◦ ,v + φ ′ ,v , where φ ◦ ,v is the standard Schwartz function and c = φ ,v (0) . Then we set φ ,v ( χ F,v ) := cφ ◦ ,v + L (1 , η v ) L (1 , η v χ F,v ) φ ′ ,v . (3.3.2)We need to show that, upon substituting it in the expression for the local Whittaker functionsgiven in Proposition 3.2.3.1, we obtain a Laurent polynomial in χ F,v ( ̟ v ) (which gives the canonicalcoordinate on Y F,v ∼ = G m,M ). By linearity and Lemma 3.2.4, it suffices to show this for thesummand L (1 ,η v ) L (1 ,η v χ F,v ) φ ′ ,v , whose coefficient is designed to cancel the factor L (1 , η v , χ v ) appearingin that expression. The only source of possible poles is the infinite sum. For n sufficiently large, if a is not in the image of uq then D n ( a ) is empty and therefore the sum is actually finite. On theother hand if a = uq ( x a ) , then for n large the function φ ,v is constant and equal to φ ,v ( x a ) on D n ( a ) ; it follows that R D n ( a ) φ ,v ( x , u ) d u x = c ′ q − nF,v for some constant c ′ independent of n and χ F,v . Then the tail of the sum is X n ≥ n c ′ χ F,v ( ̟ v ) n = c ′ χ F,v ( ̟ v ) n − χ F,v ( ̟ ) ; its product with the factor − χ F,v ( ̟ v ) appearing in front of it is then also a polynomial in χ F,v ( ̟ v ) .Finally, the last two statements of part 2 follow by the construction and Lemma 3.2.2. Proposition 3.3.2 . —
There is a bounded Y F -family of q -expansions of twisted modular formsof parallel weight E ( u, φ p ∞ ) such that for any χ F ∈ Y l . c .F ( C ) and any r = ( r v ) v | p satisfying c ( χ F ) | p r , we have E ( u, φ p ∞ ; χ F ) = | D F | L ( p ) (1 , ηχ F ) L ( p ) (1 , η ) q E r ( u, φ , χ F ) , where φ = φ p ∞ ( χ F ) φ ,p ∞ with φ ,v the standard Schwartz function for v | p ∞ .Proof . — This follows from Lemma 3.3.1 and Proposition 3.2.3.3: we take the q -expansion withcoefficients [ F : Q ] | D F | / | D E | / | L ( p ) (1 ,η ) W a ( y, u, φ p ∞ ) . We first construct certain bounded Y ? -families of q -expansions ofmodular forms, for Y ? = Y F or Y ′ . In general, if Y ? is the space of p -adic characters of aprofinite group Γ ? , then it is equivalent to give a compatible system, for each extension L ′ of L , ofbounded functionals C (Γ ? , L ′ ) → M ( K p , L ′ ) , where the source is the space of L ′ -valued continuousfunctions on Γ ? . This can be applied to the case of Y F (with Γ F ), and to the case of Y ′ with thevariation that Y ′ -families correspond to bounded functionals on the space C (Γ , ω, L ′ ) of functions f on Γ satisfying f ( zt ) = ω − ( z ) f ( t ) for all z ∈ A ∞ , × . DANIEL DISEGNI
Let φ p ∞ be a Schwartz function and let U p ⊂ B ∞× be a compact open subgroup fixing φ p ∞ . For φ ∈ S ( V ) a Schwartz function such that φ , ∞ is standard, let θ ( u, φ ) be the twisted modularform θ ( g, u, φ ) := X x ∈ E r ( g ) φ ( x , u ) . We define the modular form I F,r ( φ ⊗ φ , χ F ) = c U p | D F | / · L ( p ) (1 , ηχ F ) L ( p ) (1 , η ) X u ∈ µ Up \ F × θ ( u, φ ) E r ( u, φ , χ F ) (3.4.1)for sufficiently large r = ( r v ) v | p , and the Y F -family of q -expansions of weight modular forms I F ( φ ∞ ⊗ φ p ∞ ; χ F ) = c U p X u ∈ µ Up \ F × q θ ( u, φ ) E ( u, φ p ∞ ; χ F ) , (3.4.2)where letting µ U p = F × ∩ U p O × B ,p we set c U p := 2 [ F : Q ] − h F [ O × F : µ U p ] (3.4.3)and φ ( x , x , u ) = φ ( x , u ) φ ( x , u ) with φ i = φ p ∞ i φ i,p ∞ for φ i,v the standard Schwartz functionif v |∞ or i = 2 and v | p . The definition is independent of the choice of U p (cf. [60, (5.1.3)]).The action of the subgroup T ( A ) × T ( A ) ⊂ B × × B × on S ( V × A × ) = S ( V × A × ) ⊗ S ( V × A × ) preserves this tensor product decomposition, thus it can be written as r = r ⊗ r for theactions r , r on each of the two factors. We obtain an action of T ( A ∞ ) × T ( A ∞ ) on the forms I F,r and the families I F with orbits I F,r (( t , t ) , φ ⊗ φ , χ F ) := c U p | D F | / L ( p ) (1 , ηχ F ) L ( p ) (1 , η ) X u ∈ µ Up \ F × θ ( u, r ( t , t ) , φ ) E r ( u, φ , χ ιF ) I F (( t , t ) , φ ∞ ⊗ φ p ∞ ; χ F ) := c U p X u ∈ µ Up \ F × q θ ( q ( t ) u, r ( t , t ) φ ) E ( q ( t ) u, φ p ∞ ; χ F ) . It is a bounded action in the sense that the orbit { I F (( t , t ) , φ ∞ ⊗ φ p ∞ ) | t , t ∈ T ( A ∞ ) } isa bounded subset of the space of Y F -families of q -expansions, as both E and q θ are bounded interms of max | φ p ∞ | .Define, for the fixed finite order character ω : F × \ A × → M × , I F,ω − (( t , t ) , φ ∞ ⊗ φ p ∞ ; χ F ) := − Z A × ω − ( z ) χ F ( z ) I F (( zt , t ) , φ ∞ ⊗ φ p ∞ ; χ F ) dz, (3.4.4)a bounded Y F -family of q -expansions of forms of central character ω − , corresponding to a boundedfunctional on C (Γ F , L ) valued in M ( K p , ω − , L ) for a suitable K p .We further obtain a bounded functional I on C (Γ , ω, L ) , valued in M ( K p , ω − , L ) , which isdefined on the set (generating a dense subalgebra) of finite order characters χ ′ ∈ C (Γ , ω, L ) by I ( φ p ∞ ; χ ′ ) := Z [ T ] χ ′ ( t ) I F,ω − (( t, , φ p φ ,p ⊗ φ p ; ω · χ ′ | A × ) d ◦ t if φ p ∞ = φ p ∞ ⊗ φ p ∞ . Here φ ∞ = φ p ∞ φ ,p with φ ,v ( x , u ) = δ ,U v ( x ) O × F,v ( u ) (3.4.5)if v | p , where U p ⊂ O × E,p is a compact open subgroup small enough that χ ′ p | U ∩ T ( A ) = 1 and δ ,U v denotes the U v × U v -invariant Schwartz function with smallest support containing , having exactlytwo values, and of total mass equal to vol( O E,v , dx ) = vol( O × E,v , d × x ) for the measures dx | d | , d × x | d | on HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES E v , E × v . (The notation is meant to suggest a Dirac delta at in the variable x , to which this isan approximation).By construction, the induced rigid analytic function on Y ′ = Y ′ ω , still denoted by I , satisfiesthe following. Proposition 3.4.1 . —
There is a bounded Y ′ -family of q -expansions of modular forms I ( φ p ∞ ) such that for each χ ′ ∈ Y ′ l . c . ( C ) we have I ( φ p ∞ ; χ ′ ) = | D E | / | D F | q I r ( φ, χ ′ ι ) , where I r ( φ, χ ′ ) := Z ∗ [ T ] χ ′ ( t ) I F,r (( t, , φ, χ F ) dt with I F,r ( φ ) as in (3.4.1) , with φ p ∞ chosen as above. Let χ ′ ∈ Y ′ l . c .M ( α ) ( C ) be a character, ι : M ( α ) ֒ → C be the induced embedding. Let ψ : A /F → C × be an additive character, and let r = r ψ be theassociated Weil representation. Proposition 3.5.1 . —
Let ϕ ∈ σ ι be a form with factorisable Whittaker function, and let φ = ⊗ v φ v ∈ S ( V × A × ) . For sufficiently large r = ( r v ) v | p , we have (3.5.1) Y v | p ια ( ̟ v ) − r v · ( ϕ, w − r I r ( φ, χ ′ )) = Y v R ◦ r,v ( W v , φ v , χ ′ v , ψ v ) where R ◦ r,v ( W v , φ v , χ ′ v , ψ v ) = ια v ( ̟ v ) − r v L ( p ) (1 , η v χ F,v ) L ( p ) (1 , η v ) R r,v with R r,v = Z Z ( F v ) N ( F v ) \ GL ( F v ) W − ,v ( g ) δ χ F,r,v ( g ) Z T ( F v ) χ ′ v ( t ) r ( gw − r )Φ v ( t − , q ( t )) dt dg. Here Φ v = φ v if v is non-archimedean and Φ v is a preimage of φ v under (3.1.3) if v is archimedean, W − ,v is the local Whittaker function of ϕ for the character ψ v , and we convene that r v = 0 , w r,v = 1 , ια v ( ̟ v ) − r v = 1 if v ∤ p . Note that the integral R r,v does not depend on r ≥ unless v | p and it does not depend on χ ′ if v |∞ ; we will accordingly simplify the notation in these cases. Proof . — This is shown similarly to [60, Proposition 2.5]; se [60, (5.1.3)] for the equality betweenthe kernel functions denoted there by I ( s, χ, φ ) (similar to our c − U p I r ( φ, χ ′ ) ) and I ( s, χ, Φ) (whichintervenes in the analogue in loc. cit of the left-hand side of (3.5.1)).We will sometimes lighten a bit the notation for R ◦ v by omitting ψ v from it. Lemma 3.5.2 . —
When everything is unramified, we have R ◦ v ( W v , φ v , χ ′ v ) = L (1 / , σ E,v ⊗ χ ′ v ) ζ F,v (2) L (1 , η v ) . Proof . — With a slightly different setup, (17)
Waldspurger [56, Lemme 2, Lemme 3] showed that R v ( W v , φ v , χ ′ v ) = L (1 / , σ E,v ⊗ χ ′ v ) ζ F,v (2) L (1 , η v χ F,v ) (17) Notably, the local measures in [56] are normalised by vol( GL ( O F,v )) = 1 for almost all finite place v , whereaswe have vol( GL ( O F,v )) = ζ F,v (2) − | d | v (cf. [60, p. 23]; the second displayed formula of [60, p. 42] neglects thisdiscrepancy). DANIEL DISEGNI when χ F,v = | · | s , but his calculation goes through for any unramified character χ F,v .Define R ♮r,v ( W v , φ v χ ′ v , ψ v ) := | d v | − | D v | − / ζ F,v (2) L (1 , η v ) L (1 / , σ E,v ⊗ χ ′ v ) R ◦ r,v ( W v , φ v χ ′ v , ψ v ) . (3.5.2)Then the previous lemma combined with Proposition 3.5.1 gives: Proposition 3.5.3 . —
We have ια − r ( ϕ, w − r I ( φ, χ ′ )) = | D F | − | D E | − / L ( ∞ ) (1 / , σ E ⊗ χ ′ ) ζ ( ∞ ) F (2) L ( ∞ ) (1 , η ) Y v ∤ ∞ R ♮r,v ( W v , φ v , χ ′ v ) Y v |∞ R ◦ v ( W v , Φ v , χ ′ v ) , where all but finitely many of the factors in the infinite product are equal to .Archimedean zeta integral . — We compute the local integral R v when v |∞ . Lemma 3.5.4 . — If v |∞ , φ v is standard, and W − ,v is the standard antiholomoprhic Whittakerfunciton of weight of (2.1.2) , then R ◦ v ( W v , φ v , χ ′ v ) = R v ( W v , φ v , χ ′ v ) = 1 / . Proof . — By the Iwasawa decomposition we can uniquely write any g ∈ GL ( R ) as g = (cid:18) x (cid:19) (cid:18) z z (cid:19) (cid:18) y (cid:19) (cid:18) cos θ sin θ − sin θ cos θ (cid:19) with x ∈ R , z ∈ R × , y ∈ R × , θ ∈ [0 , π ) ; the local Tamagawa measure is then dg = dxd × z d × y | y | dθ .The integral in Z ( R ) ⊂ T ( R ) realizes the map Φ → φ ; and it is easy to verify that r ( g ) φ (1 , isthe standard holomorphic Whittaker functon of weight .We then have, dropping subscripts v : R v ( ϕ, φ ) = Z T ( R ) /Z ( R ) Z π Z R × ( | y | e − πy ) d × y | y | dθ dt = 2 · (4 π ) − π = 1 / , where (4 π ) − comes from a change of variable, T ( R ) /Z ( R )) , and π comes from theintegration in dθ . When v ∤ p , the normalised local zeta integralsadmit an interpolation as well. Recall from §1.2 that Ψ v denotes the scheme of all local additivecharacters of level . Proposition 3.6.1 . —
Let v ∤ p be a finite place, and let K ( σ v , ψ univ ,v ) be the universal Kirillovmodel of σ v . Then for any φ v ∈ S ( V v × F × v ) , W v ∈ K ( σ v , ψ univ ,v ) there exists a function R ♮ ( W v , φ v ) ∈ L (1 , η v χ F,v ) O Y ′ v × Ψ v ( Y ′ v , ω v χ − F,v, univ ) such that for all χ ′ v ∈ Y ′ v ( C ) , ψ v ∈ Ψ v ( C ) , we have R ♮v ( W v , φ v ; χ ′ v , ψ v ) = R ♮v ( W ιv , φ v ( χ ′ v ) , χ ′ v , ψ v ) , where φ v ( χ ′ v ) = φ ,v φ ,v ( χ F,v ) with φ ,v ( χ F,v ) is as in (3.3.2) . In the statement, we consider L (1 , η v χ F,v ) − as an element of O ( Y ′ v ) (coming by pullback from Y F,v ). Note that it equals the nonzero constant L (1 , η v ) − along Y v ⊂ Y ′ v . HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES Proof . — Note that the assertion on the subsheaf of O Y v × Ψ v of which R ♮v is a section simplyencodes the dependence of R ♮v on the additive character, which is easy to ascertain by a change ofvariables. By the definitions, it suffices to show that L (1 / , σ E,v ⊗ χ ′ v ) − R v ( W v , φ v , χ ′ v ) (3.6.1)extends to a regular function on Y ′ v . We will more precisely show that L (1 / , σ E,v ⊗ χ ′ v ) − is aproduct of various factors all of which extend to regular functions on Y ′ v , and that the productof some of those factors and R v ( W v , φ v , χ ′ v ) also extends to a regular function on Y ′ v . Concretely,if A ⊂ E × v / ( E × v ∩ V p ) is any finite set, then the evaluations χ ′ v ( χ ′ v ( a )) a ∈ A define a morphism ev A : Y ′ v → G Am,M , (18) so that finite sums of evaluations of characters are regular functions on Y ′ v obtained by pullback along ev A . Interpolation of R v . — Within the expression for R v , we can use the Iwasawa decomposition andnote that integration over K = GL ( O F,v ) yields a finite sum of integrals of the form (droppingsubscripts v ) (19) Z F × f ′ ( y ) Z T ( F ) χ ′ ( t ) φ ′ ( yt − , y − q ( t )) dt dy for some Schwartz functions φ ′ and elements f ′ of the Kirillov model of σ – namely, the translatesof W − and of φ v by the action of K . (More precisely, taking into account the dependence on χ ′ of φ , also products of the above integrals and of L (1 , ηχ F,v ) − can occur; the factor L (1 , ηχ F,v ) − clearly interpolates to a regular function on Y ′ v .)It is easy to see that the integral reduces to a finite sum if either W is compactly supportedor φ ′ ( · , u ) is supported away from ∈ E . It thus suffices to study the case where φ ′ ( x , u ) = O E ( x ) φ ′ F ( u ) , and f ′ belongs to the basis of the quotient space K introduced in the proof ofLemma 2.3.1. Moreover up to simple manipulations we may assume that φ F ( u ) is is close to adelta function supported at u = 1 . We distinguish three different cases. σ v is supercuspidal . — In this case K = 0 and there is nothing to prove. σ v is a special representation St( µ | · | − ) . — In this case K is spanned by f µ = µ · O F −{ } .We find that the integral is essentially (20) if there is a place w of E above v such that, for χ ′ w := χ ′ v | E × v , the character χ ′ w · µ ◦ q of E × w is ramified; and it essentially equals Y w | v (1 − χ ′ w ( ̟ w ) µ ( q ( ̟ w )) q − E,w ) − (3.6.2)otherwise. (21) In the latter case, L (1 / , σ E,v ⊗ χ ′ w ) is also equal to (3.6.2). We conclude that (3.6.1)extends to a regular function on Y ′ v . σ v is an irreducible principal series Ind( µ, µ ′ | · | − ) . — (22) The space K has dimension and f µ as above provides a nonzero element. Again the corresponding integral yields either or (3.6.2),the latter happening precisely when (3.6.2) is a factor of L (1 / , σ E ⊗ χ ) . If µ ′ = µ , then a secondbasis element is f µ ′ , for which the same discussion applies. If µ ′ = µ , then a second basis element (18) Moreover if A is sufficiently large the morphism ev A is a closed embedding. (19) See Proposition A.2.2, Lemma A.1.1 for some more detailed calculations similar to the ones of the present proof. (20)
Here we use this adverb with the precise meaning: up to addition of and multiplication by finite combination ofevaluations of χ ′ . (21) In the last expression, q is the norm of E w /F v , whereas q E,w is the cardinality of the residue field of E w . Weapologise for the near-clash of notation. (22) Here
Ind is plain (un-normalised) induction. DANIEL DISEGNI is f ′ µ ( y ) := v ( y ) µ ( y ) O F −{ } ( y ) . The integral is essentially if some χ ′ w · µ ◦ q is ramified, and Y w | v (1 − χ ′ w ( ̟ w ) µ ( q ( ̟ w )) q − E,w ) − (3.6.3)otherwise. In the latter case, L (1 / , σ E,v ⊗ χ ′ w ) equals (3.6.3) as well. Interpolation of L (1 / , σ E,v ⊗ χ ′ v ) − . — Depending only on σ v , as recalled above for each place w | v of E there exist at most two characters ν w,i w of E × w such that for all χ ′ v ∈ Y ′ v ( C ) , we canwrite L (1 / , σ E,v ⊗ χ ′ v ) − = Q ′ w,i w (1 − ν w χ ′ w ( ̟ w )) , where the product Q ′ extends over those pairs ( w, i w ) such that ν w,i w χ ′ w is unramified. We can replace the partial product by a genuine productand each of the factors by − ′ X x ∈ O × E,w / ( V p ∩ E × w ) ν w,i w ( x ) χ ′ w ( x ) · ν w,i w χ ′ w ( ̟ w ) where P ′ denotes average. This expression is the value at χ ′ v of an element of O ( Y ′ v ) , as desired. Let let M Y ′ − Y be the multiplicative part of O ( Y ′ ) f consisting of functions whose restriction to Y is invertible. (Recall that O ( Y ′ ) f ⊂ O ( Y ′ ) b is the image of ⊗ v ∤ p ∞ O ( Y ′ v ) .) Theorem 3.7.1 . —
There exists a unique function L p,α ( σ E ) ∈ O Y ′ × Ψ p ( Y ′ , ω p χ − F, univ ,p ) b [ M − Y ′ − Y ] which is algebraic on Y ′ l . c .M ( α ) × Ψ p and satisfies L p,α ( σ E )( χ ′ , ψ p ) = π F : Q ] | D F | / L ( ∞ ) (1 / , σ ιE , χ ′ ι )2 L ( ∞ ) (1 , η ) L ( ∞ ) (1 , σ ι , ad) Y v | p Z ◦ v ( χ ′ v , ψ v ) for every χ ′ ∈ Y ′ M ( α ) ( C ) inducing an embedding ι : M ( α ) ֒ → C . Here Z ◦ v is as in Theorem A.Let Y ′◦ ⊂ Y ′ be any connected component, Y ◦ := Y ∩ Y ′◦ the corresponding connectedcomponent of Y , and let B be the quaternion algebra over A ∞ determined by (1.1.1) for any(equivalently, all) points χ ∈ Y ◦ . For any ϕ p ∞ ∈ σ p ∞ and φ p ∞ ∈ S ( V p ∞ × A p ∞ , × ) , we have ℓ ϕ p ,α ( I ( φ p ∞ )) | Y ′◦ = L p,α ( σ E ) | Y ′◦ × Ψ p Y v ∤ p ∞ R ♮v ( W v , φ v ) | Y ′◦ × Ψ p (3.7.1) in O Y ′ ( Y ′◦ ) b , where both I and R ♮ are constructed using V . On the right-hand side, the product Q v ∤ p R ♮v makes sense over Y ′◦ × Ψ p by the decomposition σ ∼ = K ( σ p , Ψ p ) ⊗ K ( σ p , Ψ p ) induced bythe Whittaker functional fixed in the definition of ℓ ϕ p ,α . (23) Proof . — The definition can be given locally by taking quotients in (3.7.1) for any given ( W p ∞ , φ p ∞ ) .Note that on the right-hand side of (3.7.1), the product is finite since by Lemma 3.5.2 we have R ♮v ( W v , φ v ) = 1 identically on Y ′ if all the data are unramified. The analytic properties of L p,α ( σ E ) are then a consequence of the following claim. Let S be any finite set of places v ∤ p ∞ , containingall the ones such that either σ v is ramified or the subgroup V v ⊂ O × E,v fixed in the Introductionis not maximal. Let Y ′◦ be a connected component and B be the associated quaternion algebra.Then for each v ∈ S , there exists a finite set of pairs ( W v , φ v ) such that the locus of commonvanishing of the corresponding functions R ♮v ( W v , φ v ) | Y ◦ is empty . (23) The p -adic L -function L p,α ( σ E ) does not depend on this choice. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES We prove the claim. Let χ v ∈ Y ◦ v be any closed point, where v ∈ S and Y ◦ v ⊂ Y v is the unionof connected components corresponding to Y ◦ . By Lemma 5.1.1 below, we have R ♮ ( W v , φ v , χ v ) = Q v ( θ ψ,v ( W v , φ v ) , χ v ) , where θ ψ,v is a Shimizu lift sending σ v × S ( V v × F × ) onto π v ⊗ π ∨ v , with π v the Jacquet–Langlands transfer of σ v to B × v . By construction of B v and the result mentionedin §1.1, the functional Q v ( · , χ v ) is non-vanishing. Therefore given χ v ∈ Y ◦ v , we can find ( W v , φ v ) such that R ♮v ( W v , φ v ; χ v ) = 0 .Consider the set of all functions R ♮v ( W v , φ v ) | Y ◦ v for varying ( W v , φ v ) . As the locus of theircommon vanishing is empty, it follows by the Nullstellensatz that finitely many of them generatethe unit ideal of O ( Y ◦ v ) . (24) This completes the proof of the claim.We now move to the interpolation property. The algebraicity on Y ′ l . c .M ( α ) is clear from the def-inition just given. By ι , which we will omit from the notation below, we can identify ϕ with anantiholomorphic automorphic form ϕ ι . By the definitions and Proposition 3.5.3, we have L p,α ( σ E )( χ ′ , ψ p ) = ℓ ϕ p ,α ( I ( φ p , χ ′ )) Q v ∤ p ∞ R ♮v ( W v , φ v ; χ ′ v , ψ v )= | D F | / ζ F (2)2 L (1 , σ, ad) · | D E | / | D F | α − r ( ϕ, w − r I ( φ, χ ′ )) Q v ∤ p ∞ R ♮v ( W v , φ v , χ ′ , ψ v )= | D F | / ζ F (2)2 L (1 , σ, ad) · L ( ∞ ) (1 / , σ E , χ ′ ) ζ ( ∞ ) F (2) L ( ∞ ) (1 , η ) Y v |∞ R ◦ v ( φ v , W v , χ ′ v , ψ v ) Y v | p R ♮r,v ( φ v , W v , χ ′ v , ψ v )= ζ F, ∞ (2)2 [ F : Q ] L (1 , σ ∞ , ad) · | D F | / L ( ∞ ) (1 / , σ E , χ ′ )2 L ( ∞ ) (1 , η ) L ( ∞ ) (1 , σ, ad) Y v | p R ♮r,v ( W v , φ v , χ ′ v , ψ v ) . Here ψ p is any additive character such that ψ = ψ p ψ p ψ ∞ vanishes on F . For v |∞ , we have ζ F,v (2) /L (1 , σ v , ad) = π − / ( π − /
2) = 2 π , so that the first fraction in the last line equals π F : Q ] .The result follows.The proof is completed by the identification R ♮r,v = Z ◦ v for v | p carried out in PropositionA.2.2. p -adic heights We recall the definition and properties of p -adic heights and prove two integrality criteria forthem. The material of §§4.2-4.3 will not be used until §§8-9.For background in p -adic Hodge theory see the summary in [39, §1] and references therein. Thenotation we use is completely standard; it coincides with that of loc. cit. except that we shallprefer to write D dR instead of DR for the functor of de Rham periods. Let F be a number field and G F := Gal( F /F ) . Let L be a finite extension of Q p , and let V be a finite-dimensional L -vector space with a continuousaction of G F . For each place v of F , we denote by V v the space V considered as a representationof G F,v := Gal( F v /F v ) . Suppose that: – V is unramified outside of a finite set of primes of F ; – V v is de Rham, hence potentially semistable, for all v | p ; – H ( F v , V ) = H ( F v , V ∗ (1)) = 0 for all v ∤ p ; – D crys ( V v ) ϕ =1 = D crys ( V v ) ϕ =1 = 0 for all v | p (where ϕ denotes the crystalline Frobenius). (24) Recall that Y v is an affine scheme of finite type over M (more precisely it is a closed subscheme of a split torus). DANIEL DISEGNI
Under those conditions, Nekovář [39] (see also [41]) constructed a bilinear pairing on the Bloch–Kato Selmer groups h , i : H f ( F, V ) × H f ( F, V ∗ (1)) → Γ F ˆ ⊗ L (4.1.1)depending on choices of L -linear splittings of the Hodge filtration Fil D dR ( V v ) ⊂ D dR ( V v ) (4.1.2)for the primes v | p . In fact in [39] it is assumed that V v is semistable; we will recall the definitionsunder this assumption, and at the same time see that they can be made compatible with extendingthe ground field (in particular, to reduce the potentially semistable case to the semistable case).Cf. also [5] for a very general treatment.Post-composing h , i with a continuous homomorphism ℓ : Γ F → L ′ , for some L -vector space L ′ ,yields an L ′ -valued pairing h , i ℓ (the cases of interest to us are L ′ = L with any ℓ , or L ′ = Γ F ˆ ⊗ L with the tautological ℓ ). For such an ℓ we write ℓ v := ℓ | F × v and we say that ℓ v is unramified if itis trivial on O × F,v (note that this is automatic if v ∤ p ).Let x ∈ H f ( F, V ) , x ∈ H f ( F, V ∗ (1)) . They can be viewed as classes of extensions of Galoisrepresentations e : 0 → V → E → L → e : 0 → V ∗ (1) → E → L → . For any e , e as above, the set of Galois representations E fitting into a commutative diagram (cid:15) (cid:15) (cid:15) (cid:15) / / L (1) / / E ∗ (1) (cid:15) (cid:15) / / V / / (cid:15) (cid:15) / / L (1) / / E (cid:15) (cid:15) / / E / / (cid:15) (cid:15) L (cid:15) (cid:15) L (cid:15) (cid:15) is an H ( F, L ) -torsor; any such E is called a mixed extension of e , e ∗ (1) . Depending on the choiceof (extensions e and e and) a mixed extension E , there is a decomposition h x , x i ℓ = X v ∈ S F h x ,v , x ,v i ℓ v ,E,v (4.1.3)of the height pairing into a (convergent) sum of local symbols indexed by the non-archimedean places of F . We recall the definition of the latter [39, §7.4]. The representation E can be shownto be automatically semistable at any v | p ; for each v it then yields a class [ E v ] ∈ H ∗ ( F v , E ) with ∗ = ∅ if v ∤ p , ∗ = st if v | p . This group sits in a diagram of exact sequences(4.1.4) / / H ( F v , L (1)) / / H ∗ ( F v , E ) / / H f ( F v , V ) / / / / H f ( F v , L (1)) / / O O H f ( F v , E ) / / O O H f ( F v , V ) / / . HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES If v | p , the chosen splitting of (4.1.2) uniquely determines a splitting s v : H ∗ ( F v , E ) → H ( F v , L (1)) ;if v ∤ p , there is a canonical splitting independent of choices, also denoted by s v . In both cases, thelocal symbol is h x ,v , x ,v i ℓ v ,E,v := − ℓ v ( s v ([ E v ])) , where we still denote by ℓ v the composition H ( F v , L (1)) ∼ = F × v ˆ ⊗ L → Γ F ˆ ⊗ L → L ′ . When v | p , we say that [ E v ] is essentially crystalline if [ E v ] ∈ H f ( F v , E ) ⊂ H ( F v , E ) ; equivalently, s v ([ E v ]) ∈ H f ( F v , L (1)) . Behaviour under field extensions . — If F ′ w /F v is a finite extension of local non-archimedean fields,the pairing h , i ℓ v ◦ N Fw/Fv : H f ( F ′ w , V w ) × H f ( F ′ w , V ∗ w (1)) → L ′ (4.1.5)defined using the induced Hodge splittings and the map ℓ w := ℓ v ◦ N F w /F v satisfies h cores F ′ w F v x , x i ℓ v = h x , res F ′ w F v x i ℓ v ◦ N Fw/Fv (4.1.6)for all x ∈ H f ( F v , V v ) , x ∈ H f ( F ′ w , V | ∗ G F ′ w (1)) .Back to the global situation, it follows that extending any ℓ : Γ F → L ′ to the direct system (Γ F ′ ) F ′ /F finite by ℓ w = ℓ | F ′× w := 1[ F ′ : F ] ℓ v ◦ N F ′ w /F v (4.1.7)we can extend h , i ℓ to a pairing h , i ℓ : lim −→ F ′ H f ( F ′ , V | G F ′ ) × H f ( F ′ , V | ∗ G F ′ (1)) → L ′ , (4.1.8)where the limit is taken with respect to restriction maps. This allows to define the pairing in thepotentially semistable case as well. Ordinariness . — Let v | p be a place of F . Definition 4.1.1 . — We say that a de Rham representation V v of G F v satisfies the Panchishkincondition or that is potentially ordinary if there is a (necessarily unique) exact sequence of deRham G F v -representations → V + v → V v → V − v → with Fil D dR ( V + v ) = D dR ( V − v ) / Fil = 0 .If V v is potentially ordinary, there is a canonical splitting of (4.1.2) given by D dR ( V v ) → D dR ( V − v ) = Fil D dR ( V v ) . (4.1.9) Abelian varieties . — If
A/F is an abelian variety with potentially semistable reduction at all v | p , then the rational Tate module V = V p A satisfies the required assumptions, and there is acanonical isomorphism V ∗ (1) ∼ = V p A ∨ . Suppose that there is an embedding of a number field M ֒ → End ( A ) ; its action on V induces a decomposition V = ⊕ p | p V p indexed by the primes of O M above p . Given such a prime p , a finite extension L of M p , and splittings of the Hodge filtrationon D dR ( V p | G Fv ) ⊗ M p L for v | p , we obtain from the compatible pairings (4.1.8) a height pairing h , i : A ( F ) × A ∨ ( F ) → Γ F ˆ ⊗ L (4.1.10)via the Kummer maps κ A,F ′ , p : A ( F ′ ) → H f ( F ′ , V ) → H f ( F ′ , V p ) and κ A ∨ ,F ′ , p : A ( F ′ ) → H f ( F ′ , V ∗ p (1)) for any F ⊂ F ′ ⊂ F .If p is a prime of M above p and V p A ⊗ L is potentially ordinary for all v | p , the height pairing(4.1.10) is then canonical (cf. [41, §11.3]). Such is the situation of Theorem B. In that case we DANIEL DISEGNI consider the restriction of (4.1.10) to A ( χ ) , coming from the Kummer maps κ A ( χ ) : A ( χ ) → H f ( E, V p A ( χ )) , κ A ∨ ( χ − ) : A ∨ ( χ − ) → H f ( E, ( V p A ( χ ) ∗ )(1)) , where V p A ( χ ) := V p A | G E ⊗ L ( χ ) χ . Note that by the condition χ | A ∞ , × = ω − A , we have ( V p A ( χ )) ∗ (1) ∼ = V p A ( χ ) . If X/F is a (connected, smooth, proper) curvewith semistable reduction at all v | p , let V := H ét ( X F , Q p (1)) . Then V satisfies the relevantassumptions; moreover it carries a non-degenerate symplectic form by Poincaré duality, inducingan isomorphism V ∼ = V ∗ (1) . For any finite extension L of Q p , any Hodge splittings on ( D dR ( V v ⊗ L )) v | p , and any continuous homomorphism ℓ : Γ F → L ′ , we obtain a pairing h , i X,ℓ : Div ( X F ) × Div ( X F ) → L ′ (4.2.1)via the Kummer maps similarly to the above. The pairing factors through Div ( X F ) → J X ( F ) where J X is the Albanese variety; it corresponds to the height pairing on J X ( F ) × J ∨ X ( F ) via thecanonical autoduality of J X .The restriction of (4.2.1) to the set (Div ( X F ) × Div ( X F )) ∗ of pairs of divisors with disjointsupports admits a canonical decomposition h , i X,ℓ = X w ∈ S F ′ h , i X,ℓ w ,w . Namely, the local symbols are continuous symmetric bi-additive maps given by h D , D i X,ℓ w := h x , x i ℓ w ,E,w (4.2.2)where x i is the class of D i in H f ( F ′ , V ) , and if Z , Z ⊂ X F are disjoint proper closed subsets of X F such that the support of D i is contained in Z i , then E , E , E are the extensions obtainedfrom the diagram of étale cohomology groups (cid:15) (cid:15) (cid:15) (cid:15) / / H ( Z , L (1)) / / H (( X F , Z ) , L (1)) (cid:15) (cid:15) / / H ( X F , L (1)) / / (cid:15) (cid:15) / / H ( Z , L (1)) / / H (( X F − Z , Z ) , L (1)) (cid:15) (cid:15) / / H ( X F − Z , L (1)) / / (cid:15) (cid:15) H Z ( X F , L (1)) (cid:15) (cid:15) H Z ( X F , L (1)) (cid:15) (cid:15) by pull-back along cl D : L → H Z ( X F , L (1)) and push-out along − Tr D : H ( Z , L (1)) → L (1) .If X does not have semistable reduction at the primes above p , we can still find a finite extension F ′ /F such that X F ′ does, and define the pairing on X F ′ . If X = ` i X i is a disjoint union of finitelymany connected curves, then Div ( X F ) will denote the group of divisors having degree zero oneach connected component; it affords local and global pairings by direct sum. A uniqueness principle . — Suppose that D = div ( h ) is a principal divisor with support disjointfrom the support of D , and let h ( D ) := Q P h ( P ) n P . Then the mixed extension [ E w ] = [ E D ,D ,w ] HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES is the image of h ( D ) ⊗ ∈ F ′× w ˆ ⊗ L ∼ = H ( F ′ w , L (1)) in H ∗ ( F ′ w , E ) under (4.1.4); it follows that h D , D i X,ℓ w = ℓ w ( h ( D )) (4.2.3)independently of the choice of Hodge splittings. When ℓ w is unramified, this property in factsuffices to characterise the pairing. Lemma 4.2.1 . —
Let
X/F v be a smooth proper curve over a local field F v and suppose that ℓ v : F × v → L is unramified. Then there exists a unique locally constant symmetric biadditivepairing h , i X,ℓ v : (Div ( X F v ) × Div ( X F v )) ∗ → L such that h div ( h ) , D i X,ℓ v = ℓ v ( h ( D )) whenever the two arguments have disjoint supports.Proof . — The result is well-known, see e.g. [14, Proposition 1.2], but for the reader’s conveniencewe recall the proof. A construction of such a pairing has just been recalled, and a second one willbe given below. For the uniqueness, note that the difference of any two such pairings is a locallyconstant homomorphism J ( F v ) × J ( F v ) → L . As the source is a compact group and the target istorsion-free, such a homomorphism must be trivial. Arithmetic intersections . — Let F ′ ⊂ F be a finite extension of F and X / O F ′ be a regularintegral model of X . For a divisor D ∈ Div ( X F ′ ) , we define its flat extension to the model X to be the unique extension of D which has intersection zero with any vertical divisor; it can beuniquely written as D + V , where D is the Zariski closure of D in X and V is a vertical divisor.Let D , D ∈ Div ( X F ) be divisors with disjoint supports, with each D i defined over a finiteextension F i ; assume that F ⊂ F ⊂ F ⊂ F . Let X / O F be a regular and semistable model.Then for each finite place w ∈ S F , we can define partial local intersection multiplicities i w , j w ofthe flat extensions D + V of D , D + V of D to X O F ,w . If the latter model is still regular,they are defined by i w ( D , D ) = 1[ F : F ] ( D · D ) w ,j w ( D , D ) = 1[ F : F ] ( D · V ) w , (4.2.4)where on the right-hand sides, ( · ) w are the usual Z -valued intersection multiplicities in X O F ,w ;see [60, §7.1.7] for the generalisation of the definition to the case when X O F ,w is not regular. Thetotal intersection m w ( D , D ) = i w ( D , D ) + j w ( D , D ) is of course independent of the choice of models.Fix an extension v to F of the valuation v on F . Then we have pairings i v , j v on divisors on X F v with disjoint supports by the above formulas. We can group together the contributions of i and j according to the places of F by λ v ( D , D ) = − Z Gal(
F /F ) λ v ( D σ , D σ ) dσ for v any finite place of F and λ = i , j , or λ v ( D , D ) = h D , D i v . Here the integral uses theHaar measure of total volume one, and reduces to a finite weighted average for any fixed D , D . Proposition 4.2.2 . —
Suppose that D and D are divisors of degree zero on X , defined over anextension F ′ of F . Then for all finite places w ∤ p of F ′ , h D , D i X,ℓ w = m w ( D , D ) · ℓ w ( ̟ w ) = ( i w ( D , D ) + j w ( D , D )) · ℓ w ( ̟ w ) . DANIEL DISEGNI
Proof . — This follows from Lemma 4.2.1 and (4.2.3); the verification that the arithmetic intersec-tion pairing m w also satisfies the required properties can be found in [23]. The result of Proposition 4.2.2 applies with the same proof if w | p when ℓ w is an unramified logarithm such as the valuation. In this case we will view it as a firstintegrality criterion for local heights. Proposition 4.3.1 . —
Let ℓ v : F × v ˆ ⊗ L → Γ F ˆ ⊗ L be the tautological logarithm and let ℓ w be as in (4.1.7) . Let v : F × v ˆ ⊗ L → L be the valuation. Then for all D , D ∈ Div ( X F ′ ) we have v ( h D , D i X,ℓ w ) = [ F ′ w : F v ] · m w ( D , D ) . In particular, if m w ( D , D ) = 0 then h D , D i X,ℓ w = ℓ w ( s w ([ E D ,D ,w ])) ∈ O × F,v ˆ ⊗ L = ℓ w ( H f ( F w , L (1)); equivalently, the mixed extension [ E D ,D ,w ] is essentially crystalline. We need a finer integrality property for local heights, slightly generalising [40, Proposition1.11]. Let F v and L be finite extensions of Q p , let V be a G F v -representation on an L -vectorspace equipped with a splitting of (4.1.2), and ℓ v : F × v → L be a logarithm. Suppose the followingconditions are satisfied:(a) ℓ v : F × v → L is ramified;(b) the space V admits a direct sum decomposition V = V ′ ⊕ V ′′ as G F v -representation, such that V ′ satisfies the Panchishkin condition and the restriction of the Hodge splitting of D dR ( V ) to D dR ( V ′ ) coincides with the canonical one of (4.1.9).(c) H ( F v , V − ) = H ( F v , V + ∗ (1)) = 0 .By [39, Proposition 1.28 (3)], the last condition is equivalent to D pst ( V ′ ) ϕ =1 = D pst ( V ′∗ (1)) ϕ =1 =0 , where D pst ( V ′ ) := lim F ⊂ F ′ D st ( V ′ | G F ′ ) Gal( F ′ /F ) (the limit ranging over all sufficiently largefinite Galois extensions F ′ /F ).Let T be a G F,v -stable O L -lattice in V , T ′ := T ∩ V ′ , T ′′ = T ∩ V ′′ ; let d ≥ be an integersuch that p d L T ⊂ T ′ ⊕ T ′′ ⊂ T , where p L ∈ L is the maximal ideal of O L .Let F v ⊂ F ∞ ⊂ F ab v be the intermediate extension determined by Gal( F ab v /F v, ∞ ) ∼ = ker( ℓ v ) ⊂ F × v under the reciprocity isomorphism. Let N ∞ ,ℓ v H f ( F v , T ′ ) := \ F ′ w cores F ′ w F v ( H f ( F ′ w , T ′ )) be the subgroup of universal norms , where the intersection ranges over all finite extensions F v ⊂ F ′ v ′ contained in F v, ∞ . Proposition 4.3.2 . —
Let x ∈ H f ( F v , T ) , x ∈ H f ( F v , T ∗ (1)) and suppose that the image of x in H f ( F v , T ′′∗ (1)) vanishes. Let d be the O L -length of H ( F v , T ′′∗ (1)) tors , d the length of H f ( F v , T ′ ) /N ∞ ,ℓ v H f ( F v , T ′ ) . Then p d + d + d L h x , x i ℓ v ,E,v ⊂ ℓ v ( F × v ˆ ⊗ O L ) for any mixed extension E . If moreover [ E v ] is essentially crystalline, then p d + d + d L h x , x i ℓ v ,E,v ⊂ ℓ v ( O × F,v ˆ ⊗ O L ) . Proof . — The first assertion (which implicitly contains the assertion that H f ( F v , T ′ ) /N ∞ ,ℓ v H f ( F v , T ′ ) is finite) is identical to [40, Proposition 1.11], whose assumptions however are slightly more strin-gent. First, L is assumed to be Q p ; this requires only cosmetic changes in the proof. Secondly, in HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES loc. cit. the representation V (hence V ′ ) is further assumed to be crystalline. This assumption isused via [39, §6.6] to apply various consequences of the existence of the exact sequence → H f ( F v , V ′ + ) → H f ( F v , V ′ ) → H f ( F v , V ′− ) → (4.3.1)which is established in [39, Proposition 1.25] under the assumption that V ′ is crystalline. However(4.3.1) still exists under our assumption that H ( F v , V ′− ) = 0 by [4, Corollary 1.4.6]. (25) The second assertion follows from the proof of the first one: the height is the image under ℓ v of an element of H ( F v , L (1)) = F × v ˆ ⊗ L , which belongs to H f ( F v , L (1)) = O × F,v ˆ ⊗ L if [ E v ] isessentially crystalline.
5. Generating series and strategy of proof
We introduce the constructions that will serve to prove the main theorem. We have expressedthe p -adic L -function as the p -adic Petersson product of a form ϕ in σ A and a certain kernelfunction depending on a Schwartz function φ . The connection between the data of ( ϕ, φ ) andthe data of ( f , f ) appearing in the main theorem is given by the Shimizu lifting introduced in§5.1. An arithmetic-geoemetric analogue of the latter allows to express also the left-hand side ofTheorem B as the image under the p -adic Petersson product of another kernel function, introducedin §5.3. The main result of this section is thus the reduction of Theorem B to an identity betweenthe two kernel functions (§5.4). Let B be a quaternion algebra over a local or global field F , V = ( B, q ) with the reduced norm q . The action ( h , h ) · x := h xh − embeds ( B × × B × ) /F × inside GO ( V ) . If F is a local field, σ is a representation of GL ( F ) , and π is a representation of B × , then the space of liftings Hom GL ( F ) × B × × B × ( σ ⊗ S ( V × F × ) , π ⊗ π ∨ ) has dimension zero unless either B = M ( F ) and π = σ or σ is discrete series and π is its imageunder the Jacquet–Langlands correspondence; in the latter cases the dimension is one. An explicitgenerator was constructed by Shimizu in the global coherent case, and we can use it to normalisea generator in the local case and construct a generator in the global incoherent case. Global lifting . — Let B be a quaternion algebra over a number field F , V = ( B, q ) with thereduced norm. Let σ be a cuspidal automorphic representation of GL ( A F ) which is discreteseries at all places where B is ramified, π the automorphic representation of B × A attached to σ by the Jacquet–Langlands correspondence. Fix a nontrivial additive character ψ : A /F → C × .Consider the theta series θ ( g, h, Φ) = X u ∈ F × X x ∈ V r ψ ( g, h )Φ( x, u ) , g ∈ GL ( A ) , h ∈ ( B × A × B × A ) / A × ⊂ GO ( V A ) for Φ ∈ S ( V A × A × ) . Then the Shimizu theta lift of any ϕ ∈ σ is defined to be θ ( ϕ, Φ)( h ) := ζ F (2)2 L (1 , σ, ad) Z GL ( F ) \ GL ( A ) ϕ ( g ) θ ( g, h, Φ) dg ∈ π × π ∨ , and it is independent of the choice of ψ .If F is totally real, B is totally definite, and φ ∈ S ( V A × A × ) , we denote θ ( ϕ, φ ) := θ ( ϕ, Φ) forany O ( V ∞ ) -invariant preimage Φ of φ under (3.1.3). Let F : π ⊗ π ∨ → C be the duality definedby the Petersson bilinear pairing on B × A (for the Tamagawa measure). By [56, Proposition 5], we (25) The finiteness of H f ( F v , T ′ ) /N ∞ ,ℓ v H f ( F v , T ′ ) under our assumption is also in [41, Corollary 8.11.8]. DANIEL DISEGNI have(5.1.1) F θ ( ϕ, Φ) = ( π / [ F : Q ] | D | / F ζ ∞ F (2) Y v ∤ ∞ | d v | − / ζ F,v (2) L (1 , σ v , ad) Z N ( F ) v \ GL ( F v ) W ϕ, − ,v ( g ) r ( g )Φ v (1 , dg × Y v |∞ ζ F,v (2) π L (1 , σ v , ad) Z N ( F ) v \ GL ( F v ) W ϕ, − ,v ( g ) r ( g )Φ v (1 , dg. The terms in the first line are all rational if W φ, − ,v and Φ v are, and almost all of the factorsequal . (26) For v a real place, if W ϕ, − ,v is the standard antiholomorphic discrete series of weight and Φ v is a preimage of the standard Schwartz function φ v ∈ S ( V v × F × v ) under (3.1.2), theterms in the last line are rational too, (27) and in fact equal to by the calculation of Lemma 3.5.4. Local lifting . — In the local case, depending on the choice of ψ v , we can then normalise a generator θ v = θ ψ,v ∈ Hom GL ( F v ) × B × v × B × v ( W ( σ v , ψ v ) ⊗ S ( V v × F × v ) , π v ⊗ π ∨ v ) (where W ( σ v , ψ v ) is the Whittaker model for σ v for the conjugate character ψ v ) by F v θ v ( W, Φ) = c v ζ F,v (2) L (1 , σ v , ad) Z N ( F ) v \ GL ( F v ) W ( g ) r ( g )Φ(1 , dg (5.1.2)with c v = | d v | − / ζ F,v (2) if v is finite and c v = 2 π − if v is archimedean. Here the decompositions π = ⊗ v π v , π ∨ = ⊗ v π ∨ v are taken to satisfy F = Q v F v for the natural dualities F v : π v ⊗ π ∨ v → C .Then by (5.1.1) we have a decomposition θ = ( π / [ F : Q ] | D | / F ζ ∞ F (2) ⊗ v θ v . (5.1.3)As in the global case, we define θ v ( W, φ ) := θ v ( W, Φ) for φ = Φ ∈ S ( V v × F × v ) . Incoherent lifting . — Finally, suppose that F is totally real and B is a totally definite incoherent quaternion algebra over A = A F , and let V = ( B , q ) . Let σ be a cuspidal automorphic represen-tation of GL ( A ) which is discrete series at all places of ramification of B and let π = ⊗ v π v bethe representation of B × associated to σ by the local Jacquet–Langlands correspondence. Then(5.1.3) defines a lifting θ ∈ Hom GL ( A ) × B × × B × ( σ ⊗ S ( V × A × ) , π ⊗ π ∨ ) . It coincides with the lifting denoted by the same name in [60].
Rational liftings . — If M is a number field, σ ∞ is an M -rational cuspidal automorphic representa-tion of GL of weight as in the Introduction and π is its transfer to an M -rational representationof B ∞× under the rational Jacquet–Langlands correspondence of [60, Theorem 3.4], let σ ι be theassociated complex automorphic representation, and π ι = π ⊗ M,ι C , for any ι : M ֒ → C . Let θ ι = ⊗ v θ ιv be the liftings just constructed. Then, if B is coherent, using the algebraic Peterssonproduct of Lemma 2.4.2, there is a lifting θ : σ ∞ ⊗ S ( V ∞ × A ∞ , × ) → π ⊗ π ∨ , which is definedover M and satisfies ιθ ( ϕ, φ ∞ ) = θ ι ( ϕ ι , φ ∞ φ ι ∞ ) (5.1.4)if ϕ ι is as described before Lemma 2.4.2 and φ ι ∞ is standard. On the left hand side, we view π and π ∨ indifferently as a representation of B ∞× or B × by tensoring on each with generators of the trivialrepresentation at infinite places which pair to under the duality (this ensures compatibility withthe decomposition). After base-change to M ⊗ O Ψ v (Ψ v ) , there are local liftings at finite places (26) The analogous assertion in [60, Proposition 2.3] is incorrect. (27)
The analogous statement holds for discrete series of arbitrary weight. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES θ ψ univ ,v : W ( σ v , ψ univ ,v ) ⊗ S ( V v × F × v ) → π v ⊗ π ∨ v ⊗ O Ψ v (Ψ v ) , satisfying ιθ ψ univ ,v ( W, φ )( ψ v ) = θ ιψ,v ( W ι , φ ι ) . They induce an incoherent global lifting on σ ∞ ⊗ S ( V ∞ × A ∞ , × ) , which is definedover M independently of the choice of an additive character of A trivial on F , (28) and satisfies(5.1.4).Finally, for an embedding ι ′ : M ֒ → L with L a p -adic field, we let θ ι ′ : ( σ ∞ ⊗ S ( V ∞ × A ∞ , × )) ⊗ M L → ( π × π ∨ ) ⊗ M,ι ′ L be the base-change. Local toric periods and zeta integrals . — Recall from the Introduction that for any dual pair ofrepresentations π v ⊗ π ∨ v isomorphic (possibly after an extension of scalars) to the local componentof π ⊗ π ∨ , the normalised toric integrals of matrix coefficients of (1.1.2) are defined, for any χ ∈ Y v ( C ) , by Q v ( f ,v , f ,v , χ ) = | D | − / v | d | − / v L (1 , η v ) L (1 , π v , ad) ζ F,v (2) L (1 / , π ιE,v ⊗ χ v ) Q ♯v ( f ,v , f ,v , χ ιv ) ,Q ♯v ( f ,v , f ,v , χ v ) = Z E × v /F × v χ v ( t )( π ( t ) f ,v , f ,v ) dt. (5.1.5)The following lemma follows from the normalisation (5.1.2) and the definitions of the local toriczeta integrals R ♮r,v in (3.5.2). Lemma 5.1.1 . —
Let χ ∈ Y l . c .M ( α ) ( C ) , and ψ = ⊗ v ψ v : A /F → C be a nontrivial additive charac-ter. Let Q v and Q ♯v be the parings defined above for the representation θ ψ,v ( W ( σ v , ψ v ) ⊗ S ( V v × F × v )) . Then for all v ∤ ∞ , we have | d | − / v R ◦ r,v ( W v , φ v , χ v , ψ v ) = Q ♯v ( θ ψ,v ( W v , ια v ( ̟ v ) − r v w − r,v φ v ) , χ v ) , where as usual if v ∤ p we set α v = 1 , r v = 0 , w r,v = 1 .If v ∤ p , we have R ♮ ( W v , φ v , χ v , ψ v ) = Q v ( θ ψ,v ( W v , φ v ) , χ v ) , and for the product Q p = Q v ∤ p ∞ Q v we have Y v ∤ p ∞ R ♮r,v ( W v , φ v , χ v , ψ v ) = D / F ζ ∞ F (2)( π / [ F : Q ] Q p ( θ ψ ( ϕ, α ( ̟ ) − r w − r φ ) , χ ) . Referring to [60, §3.1] for moredetails, let us recall some basic notions on the Shimura curves X U . The set of geometricallyconnected components is π ( X U,F ) ∼ = F × + \ A ∞ , × /q ( U ) . The curve X U admits a canonical divisorclass (the Hodge class ) ξ U = L U L U of degree on each geometrically connected component;here L U is a line bundle defined over F obtained by modifying the canonical bundle. Hecke correspondences . — For x ∈ B ∞× , let T x : X xUx − → X U be the translation, given in thecomplex uniformisation by T x ([ z, y ]) = [ z, yx ] . Let p : X U ∩ xUx − → X U be the projection and let Z ( x ) U be the image of (p , p ◦ T x ) : X U ∩ xUx − → X U × X U . We view Z ( x ) U as a correspondence on X U , and we will sometimes use the same notation for theimage of Z ( x ) U in Pic ( X U × X U ) (such abuses will be made clear in what follows). (28) In the following sense. Let
Ψ := Q v Ψ v and let Ψ ◦ ⊂ Ψ be defined by Q v ψ v | F = 1 . Then the global lifting is firstdefined on σ ∞ ⊗ S ( V ∞ × A ∞ , × ) ⊗ O Ψ (Ψ) . Its restriction to σ ∞ ⊗ S ( V ∞ × A ∞ , × ) ⊗ O Ψ (Ψ ◦ ) is invariant under thehomogeneous action of O × F on Ψ ◦ and hence is the base-change of an M -linear map σ ∞ ⊗ S ( V ∞ × A ∞ , × ) → π × π ∨ . DANIEL DISEGNI
We obtain an action of the Hecke algebra H B ∞× ,U := C ∞ c ( B ∞× ) U × U of U -biinvariant functionson B ∞× by T( h ) U = X x ∈ U \ B ∞× /U h ( x ) Z ( x ) U . Note the obvious relation Z ( x ) U = T( UxU ) U . The transpose T( h ) t equals T( h t ) with h t ( x ) := h ( x − ) . It is then easy to verify that if x has trivial components away from the set of places where U is maximal, we have Z ( x ) t U = Z ( q ( x ) − ) U Z ( x ) U . (5.2.1)Finally, for any simple quotient A ′ /F of J with M ′ = End ( A ′ ) , we have a Q -linear map T alg : π A ′ ⊗ M ′ π A ′∨ → Hom ( J, J ∨ ) f ⊗ f f ∨ ◦ f . If ι : M ′ ֒ → L ′ is any embedding into a p -adic field L ′ , we denote T alg ,ι : π A ′ ⊗ M ′ π A ′∨ ⊗ M ′ L ′ ι → π A ′ ⊗ M ′ π A ′∨ ⊗ Q L ′ T alg ⊗ −→ Hom ( J, J ∨ ) ⊗ Q L ′ the composition in which the first arrow is deduced from the unique L ′ -linear embedding L ′ ֒ → M ′ ⊗ Q L ′ whose composition with M ′ ⊗ Q L ′ → L ′ , x ⊗ y ι ( x ) y , is id L ′ . Generating series . — For any φ ∈ S ( V × A × ) invariant under K = U × U , define a generatingseries Z ( φ ) := Z ( φ ) U + Z ∗ ( φ ) U where Z ( φ ) U := − X β ∈ F × + \ A ∞ , × /q ( U ) X u ∈ µ Up \ F × E ( β − u, φ ) L K,β ,Z a ( φ ) U := w U X x ∈ K \ B ∞× φ ( x, aq ( x ) − ) Z ( x ) U for a ∈ F × ,Z ∗ ( φ ) U := X a ∈ F × Z a ( φ ) U with w U = |{± } ∩ U | . Here L K,β denotes the component of a Hodge class in
Pic( X U × X U ) Q obtained from the classes L U (see [60, §3.4.4]), and the constant term E ( u, φ ) = φ (0 , u ) + W ( u, φ ) is the constant term of the standard Eisenstein series: its intertwining part W ( u, φ ) is the valueat s = 0 of W ( s, u, φ ) = Z A δ ( wn ( b )) s r ( wn ( b )) φ (0 , u ) db, where δ ( g ) = | a/d | / if g = (cid:0) a ∗ d (cid:1) k with k ∈ GL ( b O F ) SO (2 , F ∞ ) .For g ∈ GL ( A ) , define Z ( g, φ ) = Z ( r ( g ) φ ) , and similarly Z ( g, φ ) U , Z a ( g, φ ) U , Z ∗ ( g, φ ) U .Let U = U p U p and c U p be as in (3.4.3). By [60, §3.4.6], the normalised versions e Z ( g, φ ) := c U p Z ( g, φ ) U , e Z a ( g, φ ) := c U p Z a ( g, φ ) U , . . . are independent of U p . A key result, which is essentially a special case of the main theorem of[59], is that the series e Z ( g, φ ) defines an automorphic form valued in Pic ( X × X ) Q . HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES Theorem 5.2.1 . —
The map ( φ, g ) e Z ( g, φ ) defines an element e Z ∈ Hom B × × B × ( S ( V × A × ) , C ∞ ( GL ( F ) \ GL ( A )) ⊗ Pic ( X × X ) Q ) . Here the target denotes the set of
Pic ( X × X ) -valued series e Z such that for any linear functional λ : Pic ( X × X ) Q → Q , the series λ ( e Z ) is absolutely convergent and defines an automoprhic form.Its constant term is non-holomorphic in general (in fact only when F = Q and Σ = ∞ ). (Howeverthe geometric kernel that we introduce next will always be a holomorphic cusp form of weight .)See [60, Theorem 3.17, Lemma 3.18] for the proof of the theorem.Assume from now on that φ ∞ is standard; we accordingly only write Z ( φ ∞ ) , e Z ( φ ∞ ) , . . . Define,for each a ∈ A × ,(5.2.2) e Z a ( φ ∞ ) := c U p w U | a | X x ∈ K \ B ∞× φ ∞ ( x, aq ( x ) − ) Z ( x ) U for any sufficiently small U . This extends the previous definition for a ∈ F × , and it is easy tocheck that for every y ∈ A ∞ , × , a ∈ F × , we have e Z a (( y ) , φ ∞ ) = | ay | ∞ e Z ay ( φ ∞ ) ψ ( iay ∞ ) . In other words, the images in
Pic ( X U × X U ) of the e Z a ( φ ∞ ) are the reduced q -expansion coeffi-cients of e Z ( φ ∞ ) , in the following sense: for any functional λ as above, ( λ ( e Z ( y, φ )) , ( λ ( e Z a ( φ ∞ ))) a ) are the reduced q -expansion coefficients of the modular form λ ( e Z )( φ ) . Hecke operators and Hecke correspondences . — For the following lemma, let S be a set of finiteplaces of F such that for all v / ∈ S , B v is split, U v is maximal, and φ v is standard. Fix anyisomorphism γ : B S → M ( A S ) of A S -algebras carrying the reduced norm to the determinant and O B S to M ( c O F S ) ; such an isomorphism is unique up to conjugation by O × B S . Lemma 5.2.2 . —
Let U ′ S = GL ( c O F S ) , and identify the commutative algebras H SU ′ S = H S GL ( A ∞ ) ,U ′ S with H S B ∞× ,U via the isomorphism γ ∗ induced by γ above. Then for each h ∈ H SU ′ S we have T ( h ) Z ∗ ( φ ∞ ) U = T( γ ∗ h ) U ◦ Z ∗ ( φ ∞ ) U . In the left-hand side, we view Z ∗ ( φ ∞ ) as a reduced q -expansion of central character z Z ( z ) , and T ( h ) is the usual Hecke operator acting by (2.2.1) ; in the right-hand side, the symbol ◦ denotescomposition of correspondences on X U .In particular, the right-hand side is independent of the choice of γ .Proof . — It suffices to check the statements for the set of generators of the algebra H SU ′ S consistingof elements h = h v h vS , with h vS the unit of H vSU ′ S and h v = U v x v U v for x v = ( ̟ v ) or x v = (cid:16) ̟ ± v ̟ ± v (cid:17) and v / ∈ S . In the second case the statement is clear.Suppose then that x v = ( ̟ v ) . Decomposing Z a ( φ ) U = Z a v ( φ v ) U Z a v ( φ v ) U , the a th coefficientof the left-hand side equals Z a̟ v ( φ ∞ ) U + Z ( ̟ v ) U Z a/̟ v ( ϕ ∞ ) U = Z a v ( φ v ) U ◦ ( Z a v ̟ v ( φ v ) U + Z ( ̟ v ) U Z a v /̟ v ( φ v )) . It is not difficult to identify this with the a th coefficient of the right-hand side using the Cartandecomposition Z a v ( φ v ) U = X ≤ j ≤ i ≤ v ( a ) ,i + j = v ( a ) Z (cid:16)(cid:16) ̟ iv ̟ jv (cid:17)(cid:17) U (5.2.3) DANIEL DISEGNI and the relation Z (( ̟ v )) U ◦ Z (cid:16)(cid:16) ̟ iv ̟ jv (cid:17)(cid:17) U = Z (cid:16)(cid:16) ̟ i +1 v ̟ jv (cid:17)(cid:17) U + Z ( ̟ v ) U Z (cid:16)(cid:16) ̟ i − v ̟ jv (cid:17)(cid:17) U . valid whenever i > . (29) Fix a point P ∈ X E × ( E ab ) as in the Introduction, andfor any h ∈ B ∞× denote [ h ] := T h P, [ h ] ◦ = h − ξ q ( h ) , where we identify π ( X U,F ) ∼ = F × + \ A × /q ( U ) so that P U is in the component indexed by (thepoint T ( h ) P is then in the component indexed by q ( h ) , see [60, §3.1.2]). Lemma 5.3.1 . —
Fix L -linear Hodge splittings on all the abelian varieties A ′ /F parametrised by J and let h , i A ′ , ∗ be the associated local (for ∗ = v ) or global ( ∗ = ∅ ) height pairings. There areunique local and global height pairings h , i J, ∗ : J ∨ ( F ) × J ( F ) → Γ F ˆ ⊗ L such that for any A ′ and f ′ ∈ π A ′ , f ′ ∈ π ∨ A ′ , and any P ∈ J ∨ ( F ) , P ∈ J ( F ) h P , P i ∗ = h f ′∨ ◦ f ′ ( P ) , P i ∗ ,A ′ . Proof . — For each fixed level U , there is a decomposition J ∨ U ∼ ⊕ A ′ A ′∨ ⊗ π ∨ ,UA ′∨ in the isogenycategory of abelian varieties, induced by P A ′∨ ⊗ f ′∨ f ′∨ ( P A ′ ) . Then the Hodge splittings oneach A ′ induce Hodge splittings on J ∨ U . The associated pairing on J ∨ U × J U is then the unique onesatisfying the required property by the projection formula for heights (see [36]). The same formulaimplies the compatibility with respect to changing U .We consider the pairing given by the Lemma associated with arbitrary Hodge splittings on V p A ′ for A ′ = A , and any splittings on V p A ⊗ L = ⊕ p ′ V p ′ A ⊗ M p ′ L which induce the canonical one on V p A . The subscript J will be generally omitted when there is no risk of confusion.Let φ be a Schwartz function, and e Z ( φ ) be as above. Each e Z a ( φ ) gives a map e Z a ( φ ) : J ( F ) Q → J ∨ ( F ) Q by the action of Hecke correspondences. When a has trivial components at infinity and φ ∞ isstandard, we write e Z a ( φ ∞ ) := e Z a ( φ ) . Then for g ∈ GL ( A ) , h , h ∈ B ∞× we define the heightgenerating series e Z ( g, h , h , φ ∞ ) := h e Z ( g, φ )[ h ] ◦ , [ h ] ◦ i . Proposition 5.3.2 . —
The series e Z ( g, ( h , h , φ ) is well-defined independently of the choice ofthe point P . It is invariant under the left action of T ( F ) × T ( F ) and it belongs to the space ofweight cuspforms S ( K ′ , Γ F ˆ ⊗ L ) for a suitable open compact subgroup K ′ ⊂ GL ( A ∞ ) .Proof . — We explain the modularity with coefficients in the p -adic vector space Γ F ˆ ⊗ L . Supposethat U is small enough so that φ is invariant under U and U is invariant under the conjugationaction of h , h . Pick a finite abelian extension E ′ of E such that P ∈ X U ( E ′ ) and a basis { z i } of J ∨ ( E ′ ) Q , and let e i : J ∨ ( E ′ ) Q → Q be the projection onto the line spanned by z i . Then we canwrite e Z ( g, ( h , h ) , φ ∞ ) = X i h z i , [ h ] i λ i ( e Z ( g, φ ∞ )) (29) Note that a term with i = 0 only appears in (5.2.3) in the case v ( a ) = 0 , which is easily dealt with separately. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES where λ i ( T ) = e i ( T [ h ] ◦ ) . Each summand λ i ( e Z ( g, φ ∞ )) is automorphic by Theorem 5.2.1, and infact a holomorphic cuspform by [60, Lemma 3.19] (the weight can be easily computed from theshape of φ ∞ ). The other statements are proved in loc. cit. too.Define e Z ( g, φ ∞ , χ ) := Z ∗ T ( F ) \ T ( A ) /Z ( A ) χ ( t ) e Z ( g, ( t, , φ ∞ ) d ◦ t = Z ∗ T ( F ) \ T ( A ) /Z ( A ) χ ( t ) e Z ( g, (1 , t − ) , φ ∞ ) d ◦ t = Z T ( F ) \ T ( A ) /Z ( A ) χ ( t ) e Z ω − ( g, (1 , t − ) , φ ∞ ) d ◦ t where e Z ω − ( g, (1 , t − ) , φ ∞ ) = − Z Z ( A ) ω − ( z ) e Z ( g, (1 , z − t − ) , φ ∞ ) dz. Note that we have e Z ( φ ∞ , χ ) = | D E | / e Z [YZZ] ( φ ∞ , χ ) (5.3.1)if e Z [YZZ] ( φ ∞ , χ ) is the function denoted by e Z ( χ, φ ) in [60, §§ 3.6.4, 5.1.2]. Similarly to [60], we conclude thissection by reducing our main theorem to the form in which we will prove, namely as an identitybetween two kernel functions. The fundamental ingredient is the following theorem of Yuan–Zhang–Zhang.
Theorem 5.4.1 (Arithmetic theta lifting) . —
Let σ ∞ A be the M -rational automorphic repre-sentation of GL ( A ) attached to A . For any ϕ ∈ σ ∞ , we have ( ϕ, e Z ( φ ∞ )) σ ∞ = T alg ( θ ( ϕ, φ ∞ )) . in Hom (
J, J ∨ ) ⊗ M .For any ϕ ∞ ∈ σ ∞ , we have ( ϕ, e Z ( φ ∞ )) σ ∞ = | D F | T alg ( θ ( ϕ, φ ∞ )) . in Hom (
J, J ∨ ) ⊗ M .Let ι p : M ֒ → M p ⊂ L . For each ϕ p ∈ σ p ∞ A ⊗ L , completing ϕ p to a normalised (U ∗ v ) v | p -eigenform ϕ ∈ σ ⊗ L as before Proposition 2.4.4, we have ℓ ϕ p ,α ( e Z ( φ ∞ )) = | D F | T alg ( θ ι p ( ϕ, α ( ̟ ) − r w r φ ∞ )) for any sufficiently large r ≥ .Proof . — In the first identity, both sides in fact belong to M ( α ) , and the result holds if and onlyif it holds after applying any embedding ι : M ( α ) ֒ → C . It is then equivalent to [60, Theorem 3.22]via Proposition 2.4.4 and [60, Proposition 3.16]. The second identity follows from the first one andthe properties of ℓ ϕ p ,α .We can now rephrase the main theorem in the form of the following kernel identity. Theorem 5.4.2 (Kernel identity) . —
Let ϕ p ∈ σ p ∞ A and let φ p ∞ ∈ S ( V p ∞ × A p ∞ , × ) . Forany compact open subgroup U T,p = Q v U T,v ⊂ Q v | p ̟ v ) O E,p such that χ p | U T,p = 1 , let φ ∞ = φ p ∞ φ p,U T,p where φ p,U T,p = ⊗ v | p φ v,U T,v with φ v,U T,v ( x, u ) = δ ,U T,v ∩ V ( x ) O V ( x ) d − v O × F,v for δ ,U T,v as in (3.4.5) . DANIEL DISEGNI
Suppose that all primes v | p split in E . Then we have ℓ ϕ p ,α (d F I ( φ p ∞ ; χ )) = 2 | D F | L ( p ) (1 , η ) · ℓ ϕ p ,α ( e Z ( φ ∞ , χ )) . The proof will occupy most of the rest of the paper (cf. the very end of §8 below).
Proposition 5.4.3 . —
If Theorem 5.4.2 is true for some ( ϕ p , φ p ∞ ) such that for all v ∤ p ∞ thelocal integral R v ( W v , φ v , χ v ) = 0 , then it is true for all ( W p , φ p ∞ ) , and Theorem B is true for all f ∈ π , f ∈ π ∨ .Proof . — Consider the identity h T alg ,ι p ( f ⊗ f ) P χ , P χ − i J = ζ ∞ F (2)2( π / [ F : Q ] | D E | / L (1 , η ) Y v | p Z ◦ v ( α v , χ v ) − · d F L p,α ( σ A,E )( χ ) · Q ( f , f , χ ) (5.4.1)where ι p : M ֒ → L ( χ ) , and we set P χ = − Z [ T ] T t ( P − ξ P ) χ ( t ) dt ∈ J ( F ) L ( χ ) . The identity (5.4.1) is equivalent to Theorem B by Lemma 5.3.1, but it has the advantage ofmaking sense, by linearity, for any element of π ⊗ π ∨ . By the multiplicity one result, it suffices toprove it for a single element of this space which is not annihilated by the functional Q ( · , χ ) . Suchelement will arise as a Shimizu lift. (The similar assertion on the validity of Theorem 5.4.2 for all ( ϕ p , φ p ∞ ) follows from the uniqueness of the Shimizu lifting.)By (3.7.1), we can write ℓ ϕ p ,α (d F I ( φ p ∞ ; χ )) = d F L p,α ( σ E )( χ ) Y v ∤ p ∞ R ♮v ( W v , φ v ; χ v ) (note that as the functional ℓ ϕ p ,α is bounded, we can interchange it with the differentiation; thefact that the Leibniz rule does not introduce other terms follows from the vanishing of I ( φ p ∞ ; χ ) ,which will be shown in Proposition 7.1.1.3 below). By Lemma 5.1.1, this equals | D F | / ζ ∞ F (2)( π / [ F : Q ] Y v | p Q v ( θ v ( W v , α ( ̟ v ) − r v v w − r,v φ v ) , χ v ) − · d F L p,α ( σ E )( χ ) · Q ( θ ι p ( ϕ, α ( ̟ ) − r w − r φ ) , χ ) . For the geometric kernel, by Theorem 5.4.1 and the calculation of [60, §3.6.4], we have ℓ ϕ p ,α ( e Z ( φ ∞ , χ )) = 2 | D F | / | D E | / L (1 , η ) h T alg ,ι p ( θ ι p ( ϕ, α ( ̟ ) − r w − r φ )) P χ , P − χ i . Then (5.4.1) follows from Theorem 5.4.2 provided we show that, for all v | p , Q v ( θ v ( W v , α v ( ̟ v ) − r v w − r,v φ v ) , χ v ) = L (1 , η v ) − · Z ◦ v ( χ v ) . This is proved by explicit computation in Proposition A.3.1.
6. Local assumptions
We list here the local assumptions which simplify the computations, while implying the desiredidentity in general. We recall on the other hand the essential assumption, valid until the end ofthis paper, that all primes v | p split in E .Let S F be the set of finite places of F . We partition it as S F = S non-split ∪ S split with the obvious meaning according to the behaviour in E , and further as S F = S p ∪ S ∪ S ∪ ( S non-split − S ) ∪ ( S split − S p − S ) , HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES where: – S p ⊂ S split is the set of places above p ; – S is a finite subset of S non-split containing all places where E/F or F/ Q is ramified, or σ isnot an unramified principal series, or χ is ramified, or B is ramified; we assume that | S | ≥ ; – S consists of two places in S split − S p at which σ and χ are unramified.We further denote by S ∞ the set of archimedean places of F . p . — Consider the following assumptions from [60, §5.2].
Assumption 6.1.1 ( cf. [60, Assumption 5.2] ) . — The Schwartz function φ = ⊗ φ v ∈ S ( B × A × ) is a pure tensor, φ v is standard for any v ∈ S ∞ , and φ v has values in Q for any v ∈ S F . Assumption 6.1.2 ( [60, Assumption 5.3] ) . — For all v ∈ S , φ v satisfies φ v ( x, u ) = 0 if v ( uq ( x )) ≥ − v ( d v ) or v ( uq ( x )) ≥ − v ( d v ) . Assumption 6.1.3 ( [60, Assumption 5.4] ) . — For all v ∈ S , φ v satisfies r ( g ) φ v (0 , u ) = 0 for all g ∈ GL ( F v ) , u ∈ F × v . See [60, Lemma 5.10] for an equivalent condition.
Assumption 6.1.4 ( [60, Assumption 5.5] ) . — For all v ∈ S non-split − S , φ v is the standardSchwartz function φ v ( x, u ) = O B v ( x ) d − v O × F,v ( u ) . Assumption 6.1.5 ( cf. [60, Assumption 5.6] ) . — The open compact subgroup U p = Q v ∤ p U v ⊂ B ( A p ∞ ) satisfies the following:i. U v is of the form (1 + ̟ rv O B v ) × for some r ≥ ;ii. χ is invariant under U pT := U p ∩ T ( A p ∞ ) ;iii. φ is invariant under K = U p × U p ;iv. U v is maximal for all v ∈ S non-split − S and all v ∈ S ;v. U p U ,p does not contain − ;vi. U p U ,p is sufficiently small so that each connected component of the complex points of theShimura curve X U is an unramified quotient of H under the complex uniformisation.Here we have denoted by U ,p ⊂ B × p the maximal compact subgroup. See [60, §5.2.1] for an introductory discussion of the effect of those assumptions.
Lemma 6.1.6 . —
For each v ∤ p ∞ , there exist W v ∈ σ v and a Schwartz function φ v satisfyingall of Assumptions 6.1.1–6.1.5 such that R ♮v ( W v , φ v ) = 0 . For all but finitely many places v , we can take W v to be an unramified vector and φ v to be thestandard Schwartz function.Proof . — The existence of the sought-for pairs ( W v , φ v ) is proved in [60, Proposition 5.8]. Thesecond assertion follows from the unramified calculation Lemma 3.5.2. DANIEL DISEGNI p . — We make some further assumptions at the places v ∈ S p . Afterstating the restrictions on φ v and U v , we will impose at the end of this section some restrictionson χ v and on choices of a p -adic logarithm.Concerning ( φ v , U v ) , we need two conditions. On the one hand, that the centre of the opencompact subgroup U v is sufficiently large so that, roughly speaking, for all but finitely many characters χ , no nonzero vector in a T ( F v ) -representation can be both χ v -isotypic and invariantunder U T,v := U ∩ T ( F v ) ; we will apply this in particular for the space generated by the Hodgeclasses on X U . On the other hand, we need φ v to be sufficiently close (a condition depending on χ v ) to a Dirac delta, so as to match the Schwartz functions used in the construction of the p -adicanalytic kernel. As stated, this is apparently incompatible with the previous condition. However,as χ v | F × v = ω − v (fixed), an agreeable compromise can be found. We therefore state two distinctassumptions; while we will eventually work with the second assumption (the “compromise”), it willbe convenient to reduce some proofs to the situation of the first one. Assumption 6.2.1 . —
For each v ∈ S p , the subgroup U v = 1 + ̟ r v v O B v for some r v ≥ ,it satisfies ii. of Assumption 6.1.5, and the condition that α v is invariant under q ( U v ) . TheSchwartz function is φ v ( x, u ) = δ ,U T,v ∩ V ( x ) O V ( x ) δ q ( U ) ( u ) where δ ,U T,v denotes the U T,v × U T,v -invariant Schwartz function supported on U v ∩ V , v (= U T,v ) ,having exactly two values, and of total mass equal to vol( O E,v , dx ) for the Haar measures dx on E v giving vol( O E,v ) = | D | / ; (30) similarly δ q ( U ) is the finest q ( U ) -invariant approximation to adelta function at ∈ F × v , of total mass equal to vol( O × F v ) for the standard measure on F × v . Assumption 6.2.2 . —
For each v ∈ S p , the subgroup U v and the Schwartz function φ v satisfy:i. U v = U ◦ F,v e U v with U ◦ F,v = (1 + ̟ n v v O F,v ) × ⊂ Z ( F v ) ⊂ B × v for some n v ≥ , and e U v =1 + ̟ r v v O B ,v for some r v ≥ as in Assumption 6.2.1;ii. q ( U v ) ⊂ ( U ◦ F,v ) ;iii. the Schwartz function φ v = φ ω,v is φ ω,v = − Z O × F,v ω ( z ) r (( z, e φ v dz (6.2.1) where e φ v is as in Assumption 6.2.1 for e U v ;iv. U v satisfies ii.-iii. of Assumption 6.1.5 for φ ω,v , and the condition that α v is invariant under q ( U v ) . Remark 6.2.3 . — As χ v | F × v = ω − v , the subgroup U ◦ F,v in Assumption 6.2.2 can be chosen inde-pendently of χ .In view of the previous remark, we can introduce the following assumption after fixing U ◦ F,v . Assumption 6.2.4 . —
The character χ is not invariant under V ◦ p := Q v | p q − ( U ◦ F,v ) ⊂ O × E,p . Lemma 6.2.5 . —
The set of finite order characters χ ∈ Y which do not satisfy Assumption 6.2.4is finite.Proof . — Recall that by definition Y = Y ω ( V p ) parametrises some V p -invariant characters forthe open compact subgroup V p ⊂ E × A p ∞ fixed (arbitrarily) in the Introduction. Then a character χ as in the Lemma factors through E × A ∞ /E × V p V ◦ p , a finite group. (30) Cf. (3.4.5). Note that this measure differs from the standard measure on E v by | d | . HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES p -adic logarithms . — Recall that a p -adic logarithm valued in a finite extension L of Q p is acontinuous homomorphism ℓ : Γ F → L ; we call it ramified if for all v | p the restriction ℓ v := ℓ | F × v is ramified, i.e. nontrivial on O × F,v . Lemma 6.2.6 . —
For any finite extension L of Q p , the vector space of continuous homomor-phisms Hom (Γ F , L ) admits a basis consisting of ramified logarithms.Proof . — If F = Q then Hom (Γ Q , L ) is one-dimensional with generator the cyclotomic logarithm ℓ Q , which is ramified. For general F , ℓ Q ◦ N F/ Q : Γ F → Γ Q → Q p is ramified (and it generates Hom (Γ F , L ) if the Leopoldt conjecture for F holds). Any other logarithm ℓ can be written as ℓ = aℓ Q ◦ N F/ Q + ( ℓ − aℓ Q ◦ N F/ Q ) for any a ∈ L ; for all but finitely many values of a , bothsummands are ramified.
7. Derivative of the analytic kernel
For this section, we retain all the notation of §§3.2-3.4, and we keep the assumption that V isincoherent. We assume that all v | p split in E . We start by studying the incoherentEisenstein series E . Proposition 7.1.1 . —
1. Let a ∈ F × v .(a) If a is not represented by ( V ,v , uq ) then W ◦ a,v ( g, u, ) = 0 .(b) (Local Siegel–Weil formula.) If there exists x a ∈ V ,v such that uq ( x a ) = a , then W ◦ a,v ( (cid:0) y (cid:1) , u, ) = Z E v r ( (cid:0) y (cid:1) , h ) φ ,v ( x a , u ) dh
2. For any a , u ∈ F × , there is a place v ∤ p of F such that a is not represented by ( V , uq ) .3. For any φ p ∞ ∈ S ( V p ∞ × A p ∞ , × ) , u ∈ F × , we have E ( u, φ p ∞ ; ) = 0 and consequently I F ( φ p ∞ ; ) = 0 , I ( φ p ∞ ; χ ) = 0 for any φ p ∞ ∈ S ( V p ∞ × A p ∞ , × ) , χ ∈ Y ω .Proof . — Part 1 is [60, Proposition 6.1] rewritten in our normalisation – except for (b) when v | p ,which is verified by explicit computation of both sides (recall that φ ,v is standard when v | p ). Part2 is a crucial consequence of the incoherence, proved in [60, Lemma 6.3]. In view of the expansionof Proposition 3.2.1, the vanishing is a consequence of the vanishing of the nonzero Whittakerfunctions (which is implied by the previous local results) and of W ( u, ) = − L ( p ) (0 , η ) Y v W ◦ ,v ( u, ) : here we have L ( p ) (0 , η ) = L (0 , η ) Q v | p L (0 , η v ) = 0 as L (0 , η ) is defined and nonzero whereas L ( s, η v ) has a pole at s = 0 when v splits in E . DANIEL DISEGNI
Fix henceforth a tangent vector ℓ ∈ Hom (Γ F , L ( χ )) ∼ = T Y F ⊗ L ( χ ) ∼ = N ∗ Y / Y ′ | χ ; we assume that ℓ is ramified when viewed as a p -adic logarithm (cf.Lemma 6.2.6). For any function f on Y F , we denote f ′ ( ) = D ℓ f ( ) the corresponding directional derivative.Our goal is to compute, for any locally constant χ , the derivative I ′ ( φ p ∞ ; χ ) = Z ∗ [ T ] χ ( t ) I ′ F (( t, , φ p ∞ ; ) U d ◦ t = Z ∗ [ T ] χ ( t ) I ′ F ((1 , t − ) , φ p ∞ ; ) U d ◦ t, where the first identity (of q -expansions) follows from the vanishing of the values I F ( φ p ∞ ; ) .We can decompose the derivative into a sum of q -expansions indexed by the non-split finiteplaces v . For each u ∈ F × and each place v of F , let F u ( v ) be the set of those a ∈ F × representedby ( V v , uq ) ; by Proposition 7.1.1 we have W ◦ a,v ( u, ) = 0 for each a ∈ F u ( v ) , and moreover F u ( v ) is always empty if v splits in E .Then E ′ ( u ; ) = 2 [ F : Q ] | D F | / | D E | / L ( p ) (1 , η ) W ′ ( u ; ) − [ F : Q ] | D | / | D E | / L ( p ) (1 , η ) X v non-split a ∈ F u ( v ) W ◦ a,v ′ ( u ; ) W ◦ ,va ( u ; ) q a For a non-split finite place v , let E ′ ( u, φ p ∞ ; )( v ) := − [ F : Q ] | D F | / | D E/F | / L ( p ) (1 , η ) X a ∈ F u ( v ) W ′ a,v ( u ; ) W ◦ ,va ( u ; ) q a , I ′ F (( t , t ) , φ p ∞ ; )( v ) := c U p X u ∈ µ Up \ F × θ ( u, r ( t , t ) φ ) E ′ ( u, φ p ∞ ; )( v ) , I ′ ( φ p ∞ ; χ )( v ) := Z ∗ [ T ] χ ( t ) I ′ F ((1 , t − ) , φ p ∞ ; )( v ) d ◦ t if φ p ∞ = φ p ∞ ⊗ φ p ∞ , with φ obtained from φ p ∞ as in (3.4.5), and extended by linearity in general. Proposition 7.2.1 . —
Under Assumption 6.1.2, we have I ′ F ( φ p ∞ ; ) = X v non-split I ′ F ( φ p ∞ ; )( v ) . Proof . — By the definitions, we only need to show that under our assumptions we have W ′ ( u ; ) = 0 . This is proved similarly to [60, Proposition 6.7].
We give explicit expressions for the local componentsat good places, and identify the local components at bad places with certain coherent theta seriescoming from nearby quaternion algebras B ( v ) ; these theta series will be orthogonal to all forms in σ by the Waldspurger formula and the local dichotomy. Proposition 7.3.1 . —
Let v be a finite place non-split in E . Then for any ( t , t ) ∈ T ( A ) , wehave I ′ F (( t , t ) , φ p ∞ ; )( v ) = 2 | D F | L ( p ) (1 , η ) − Z [ T ] K ( v ) φ p ∞ (( tt , tt )) dt HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES and I ′ ( φ p ∞ ; χ )( v ) = 2 | D F | L ( p ) (1 , η ) Z ∗ [ T ] − Z [ T ] K ( v ) φ p ∞ (( t, tt − )) dt d ◦ t where K ( v ) φ ( y, ( t , t )) = K ( v ) r ( t ,t ) φ ( y )= c U p X u ∈ µ Up \ F × X x ∈ V − V k r ( t ,t ) φ v ( y, x, u ) r (( y ) , ( t , t )) φ v ∞ ( x, u ) q uq ( x ) with k φ v ( y, x, u ) the linear function in φ v given when φ = φ ,v ⊗ φ ,v by k φ v ( y, x, u ) := − | d | / v | D | / v vol( E v ) r (( y v )) φ ,v ( x , u ) W ◦ uq ( y ) ,v ′ ( y, u, φ ,v ) . Proof . — This follows from the definitions and the Siegel–Weil formula (Proposition 7.1.1.1b).The computation is as in [60, Proposition 6.5].
Lemma 7.3.2 . —
Assume that φ ∞ is Q -valued. For each non-split finite place v , the values ofthe function k ♮φ v ( y, x, u ) := ℓ ( ̟ v ) − k φ v ( y, x, u ) and the coefficients of the reduced q -expansions K ( v ) ♮φ p ∞ := ℓ ( ̟ v ) − K ( v ) φ p ∞ , I ′ F ♮ ( φ p ∞ )( v ) := ℓ ( ̟ v ) − I ′ F ( φ p ∞ )( v ) belong to Q .Proof . — By Lemma 3.3.1, the local Whittaker function W ◦ a,v ( y, u, φ ,v ; χ F ) belongs to O ( Y F,v ) ∼ = M [ X ± v ] and actually to Q [ X ± v ] , where X v ( χ F,v ) := χ F,v ( ̟ v ) for any uniformiser ̟ v . (Recallthat the scheme Y F,v of (3.3.1) parametrises unramified characters of F × v .) Therefore its derivativein the direction ℓ is a rational multiple of D ℓ X v = ℓ ( ̟ v ) .The following is the main result of this section. It is the direct analogue of [60, Proposition 6.8,Corollary 6.9] and it is proved in the same way, using Proposition 3.2.3.1. To compare signs with[60], note that in Proposition 7.3.1 we have preferred to place the minus sign in the definition of k φ v ; and that our ℓ ( ̟ v ) , which is the derivative at s = 0 of χ F ( ̟ v ) s , should be compared with − log q F,v in [60] (denoted by − log N v there), which is the derivative at s = 0 of | ̟ v | s . Proposition 7.3.3 . —
Let v be a non-split finite place of F , and let B v be the quaternion algebraover F v which is not isomorphic to B v .1. If v ∈ S non-split − S , then k ♮φ v (1 , x, u ) = O Bv × O × Fv ( x, u ) v ( q ( x )) + 12 .
2. If v ∈ S and φ v satisfies Assumption 6.1.2, then k ♮φ v ( y, x, u ) extends to a rational Schwartzfunction of ( x, u ) ∈ B v × F × v , and we have the identity of q -expansions K ( v ) ♮φ (( t , t )) = q θ (( t , t ) , k ♮φ v ⊗ φ v ) , where for any φ ′ , θ ( g, ( t , t ) , φ ′ ) = c U p X u ∈ µ Up \ F × X x ∈ V r ( g, ( t , t )) φ ′ ( x, u ) is the usual theta series. DANIEL DISEGNI
8. Decomposition of the geometric kernel and comparison
We establish a decomposition of the geometric kernel according to the places of F , and compareits local terms away from p with the corresponding local terms in the expansion of the analytickernel. Together with a result on the local components of the gemetric kernel at p proved in §9,this proves the kernel identity of Theorem 5.4.2 (hence Theorem B) when χ satisfies Assumption6.2.4. Fix a level U as in Assumptions6.1.5 and 6.2.2.Recall the height generating series e Z (( t , t ) , φ ∞ ) = h e Z ∗ ( φ ∞ )( t − ξ q ( t ) ) , t − ξ q ( t ) i , and the geometric kernel function e Z ( φ ∞ , χ ) = Z ∗ [ T ] χ ( t ) e Z ((1 , t − ) , φ ∞ ) dt. They are modular cuspforms with coefficients in Γ F ˆ ⊗ L ( χ ) . Proposition 8.1.1 . —
1. If Assumption 6.1.3 is satisfied, then deg e Z ( φ ∞ ) U,α = 0 for all α ∈ F × + \ A × /q ( U ) .2. If Assumption 6.1.3 is satisfied, then e Z ( φ ∞ ) ξ α = 0 for all α ∈ F × + \ A × /q ( U ) .3. If Assumption 6.2.4 is satisfied, then Z ∗ [ T ] χ ( t ) ξ U,q ( t ) dt = 0 .
4. If Assumptions 6.1.3 and 6.2.4 are both satisfied, then q e Z ( φ ∞ , χ ) U = h q e Z ∗ ( φ ∞ )1 , t χ i , where t χ = Z ∗ [ T ] χ ( t )[ t − ] U d ◦ t ∈ Div ( X U ) L ( χ ) . Proof . — Note first that part 4 follows from parts 2 and 3 after expanding and bringing theintegration inside. Part 2 follows from Part 1 as in [60, §7.3.1], and part 1 is proved in [60, Lemma7.6].For part 3, note first that the group V ◦ p of Assumption 6.2.4 acts trivially on the Hodge classes;in fact we have r (1 , t − ) ξ U,α = ξ U,αq ( t ) , and by definition q ( V ◦ p ) ⊂ U . On the other hand we areassuming that the character χ is nontrivial on V ◦ p . It follows that the integration against χ on V ◦ p ⊂ T ( A ) annihilates the Hodge classes. Let ℓ : Γ F → L ( χ ) be the ramified logarithm fixed in §7.2. For the rest of this section and in §9, we will abuse notationby writing e Z ( φ ∞ , χ ) for the image of e Z ( φ ∞ , χ ) under ℓ : Γ F ˆ ⊗ L ( χ ) → L ( χ ) . Lemma 8.2.1 . —
If Assumption 6.1.2 is satisfied, then for all a ∈ A S ∞ , × and for all t , t ∈ T ( A ∞ ) , the support of Z a ( φ ∞ ) t does not contain [ t ] . HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES Proof . — This is shown in [60, §7.2.2].Let S ′ = S ′ S ( L ( χ )) be the quotient space, relative to the set of primes S , introduced after theApproximation Lemma 2.1.2. Proposition 8.2.2 . —
Suppose that Assumptions 6.1.2, 6.1.3 and 6.2.4 are satisfied. If H is anysufficiently large finite extension of E and w is a place of H , let h , i ℓ,w be the pairing on Div ( X U,H ) associated with ℓ w of (4.1.7) . Let q e Z ∗ ( φ ∞ ) be the image of e Z ∗ ( φ ∞ ) in S ′ ⊗ Corr ( X × X ) Q . Thenin S ′ we have the decomposition q e Z ( φ ∞ , χ ) = X v e Z ( φ ∞ , χ )( v ) where e Z ( φ ∞ , χ )( v ) = X w | v h q e Z ∗ ( φ ∞ )1 , t χ i ℓ,w . Proof . — By Proposition 8.1.1.4, we have e Z a ( φ ∞ , χ ) = h e Z a ( φ ∞ )1 , t χ i for all a ∈ A ∞ , × . If a ∈ A S ∞ , × , the two divisors have disjoint supports by Lemma 8.2.1, and we can decompose theirlocal height according to (4.2.2).For each place w of E , fix an extension w of w to F ⊃ E , and for each finite extension H ⊂ F of E , let h , i w be the pairing associated with ℓ w := H w : F v ] ℓ v ◦ N H w /F v . The absence of the field H from the notation is justified by the compatibility deriving from (4.1.6). By the explicit descriptionof the Galois action on CM points we have e Z ( φ ∞ , χ )( v ) = 1 | S E v | X w ∈ S Ev − Z [ T ] h q e Z ∗ ( φ ∞ ) t, tt χ i w dt (8.2.1)in S ′ , and if v ∤ p , by Propositon 4.2.2 we can further write(8.2.2) e Z ( φ ∞ , χ )( v ) = ℓ ( ̟ v ) | S E v | X w ∈ S Ev Z ∗ [ T ] − Z [ T ] i w ( q e Z ∗ ( φ ) t, tt − ) χ ( t ) dtd ◦ t + Z ∗ [ T ] − Z [ T ] j w ( q e Z ∗ ( φ ) t, tt − ) χ ( t ) dtd ◦ t . Recall that we want to show the kernel identity ℓ ϕ p ,α ( I ′ ( φ p ∞ ; χ )) = 2 L ( p ) (1 , η ) ℓ ϕ p ,α ( e Z ( φ ∞ , χ )) (8.3.1)of Theorem 5.4.2 (more precisely we have here projected both sides of that identity to L ( χ ) via ℓ ).Similarly to Proposition 8.2.2, we have by Propositions 7.2.1, 7.3.1 a decomposition of reduced q -expansions I ′ ( φ p ∞ ; χ ) = X v non-split I ′ ( φ p ∞ ; χ )( v ) with I ′ ( φ p ∞ ; χ )( v ) = 2 | D F | L ( p ) (1 , η ) Z ∗ [ T ] − Z [ T ] K ( v ) φ p ∞ ( t, tt − ) χ ( t ) dt d ◦ t , and the q -expansion K ( v ) ♮φ p ∞ = K ( v ) φ p ∞ · ℓ ( ̟ v ) − has rational coefficients.We thus state the main theorem on the local components of the kernel function from which theidentity (8.3.1) will follow, preceded by a result on the components away from p which facilitatesthe comparison with [60]. DANIEL DISEGNI
Proposition 8.3.1 . —
Suppose that all of the assumptions of §6.1 are satisfied together withAssumptions 6.2.2. Then for all t , t ∈ T ( A ) we have the following identities of reduced q -expansions in S ′ :1. If v ∈ S split − S p , then i v ( q e Z ∗ ( φ ∞ ) t , t ) = j v ( q e Z ∗ ( φ ∞ ) t , t ) = 0 .
2. If v ∈ S non-split − S , then i v ( q e Z ∗ ( φ ∞ ) t , t ) = K ( v ) ♮φ p ∞ ( t , t ) , j v ( q e Z ∗ ( φ ∞ ) t , t ) = 0 .
3. If v ∈ S , then there exist Schwartz functions k φ v , m φ v , l φ v ∈ S ( B ( v ) v × F × v ) depending on φ v and U v such that: K ( v ) ♮φ p ∞ ( t , t ) = q θ (( t , t ) , k φ v ⊗ φ v ) ,i v ( q e Z ∗ ( φ ∞ ) t , t ) = q θ (( t , t ) , m φ v ⊗ φ v ) .j v ( q e Z ∗ ( φ ∞ ) t , t ) = q θ (( t , t ) , l φ v ⊗ φ v ) . Here B ( v ) is the coherent nearby quaternion algebra to B obtained by changing invariants at v , and for φ ′ ∈ S ( B ( v ) A × A × ) , we have the automorphic theta series θ ( g, ( t , t ) , φ ′ ) = c U p X u ∈ µ Up \ F × X x ∈ B ( v ) r ( g, ( t , t )) φ ′ ( x, u ) . We denote by I ( φ ′ , χ )( g ) := Z ∗ [ T ] − Z [ T ] χ ( t ) θ ( g, ( t, t − t ) , φ ′ ) dt d ◦ t (8.3.2)the associated coherent theta function. Proof . — Part 1 is [60, Theorem 7.8 (1)]. In part 2, the vanishing of j v follows by the definitions;the other identity is obtained by explicit computation of both sides as in [60, Proposition 8.8], whichgives the expression for the geometric side; (31) on the analytic side we use the result of Proposition7.3.3.1. Part 3 for K ( v ) ♮φ p ∞ is Proposition 7.3.3.2, whereas for i v and j v it is [60, Theorem 7.8 (4)]. Theorem 8.3.2 . —
Suppose that all of the assumptions of §6.1 are satisfied together with As-sumptions 6.2.2 and 6.2.4. Then we have the following identities of reduced q -expansions in S ′ :1. If v ∈ S split − S p , then e Z ( φ ∞ , χ )( v ) = 0 .
2. If v ∈ S non-split − S , then I ′ ( φ p ∞ ; χ )( v ) = 2 | D F | L ( p ) (1 , η ) e Z ( φ ∞ , χ )( v ) .
3. If v ∈ S , then there exist Schwartz functions k φ v , n φ v ∈ S ( B ( v )) v × F × v ) depending on φ v and U v such that, with the notation (8.3.2) : I ′ ( φ p ∞ ; χ )( v ) = q I ( k φ v ⊗ φ v , χ ) , e Z ( φ ∞ , χ )( v ) = q I ( n φ v ⊗ φ v , χ )
4. The sum e Z ( φ ∞ , χ )( p ) := X v ∈ S p e Z ( φ ∞ , χ )( v ) (31) Recall that, on the geometric side, i v is the same Q -valued intersection multiplicity both in [60] and in our case. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES belongs to the isomorphic image S ⊂ S ′ of the space of p -adic modular forms S , and we have ℓ ϕ p ,α ( e Z ( φ ∞ , χ )( p )) = 0 . Proof (to be completed in §9) . — Parts 1–3 follow from Proposition 8.3.1 by integration. Theyimply the identity in S ′ (8.3.3) | D F | L ( p ) (1 , η ) e Z ( φ ∞ , χ )( p ) = 2 | D F | L ( p ) (1 , η ) q e Z ( φ ∞ , χ ) − X v ∤ p e Z ( φ ∞ , χ )( v )= 2 | D F | L ( p ) (1 , η ) q e Z ( φ ∞ , χ ) − I ′ ( φ ∞ ; χ ) − X v ∈ S q I ( φ v ⊗ d φ v ) where d φ v = 2 | D F | L ( p ) (1 , η ) n φ v − k φ v for v ∈ S . As all terms in the right-hand side belong to S ,so does e Z ( φ ∞ , χ )( p ) . The proof of the vanishing statement of part 4 will be given in §9. Proof of Theorem 5.4.2 under Assumption 6.2.4 . — We show that Theorem 5.4.2 follows fromTheorem 8.3.2, under the same assumptions. By Proposition 5.4.3 and Lemma 6.1.6, only theAssumption 6.2.4 on the character χ is restrictive. Moreover by Lemma 6.2.6 it is equivalentto show that the desired kernel identity holds after applying to both sides a ramified logarithm ℓ : Γ F → L ( χ ) .By (8.3.3), ℓ ϕ p ,α (cid:16) | D F | L (1 , η ) · q e Z ( φ ∞ , χ ) − I ′ ( φ ∞ ; χ ) (cid:17) = X v ∈ S ℓ ϕ p ,α ( q I ( φ v ⊗ d φ v )) + ℓ ϕ p ,α ( e Z ( φ ∞ , χ )( p )) . The vanishing of the terms indexed by S can be shown as in [60, §7.4.3] to follow from the localresult of Tunnell and Saito together with Waldspurger’s formula. The term ℓ ϕ p ,α ( e Z ( φ ∞ , χ )( p )) = 0 by part 4 of Theorem 8.3.2.
9. Local heights at p After some preparation in §9.1, in §9.2 we prove the vanishing statement of part 4 of Theorem8.3.2. We follow a strategy of Nekovář [40] and Shnidman [52].For each v | p , fix isomorphisms E v := E ⊗ F F v ∼ = F v ⊕ F v (9.0.4)and B v ∼ = M ( F v ) such that the embedding of quadratic spaces E v ֒ → B v is identified with ( a, d ) ( a d ) ; then for the decomposition B v = V ,v ⊕ V ,v = E v ⊕ E ⊥ v , the first (resp. second)factor consists of the diagonal (resp. anti-diagonal) matrices. Let w , w ∗ be the places of E above v such that E w (resp. E w ∗ ) corresponds to the projection onto the first (resp. second) factor under(9.0.4). We fix the extension v = w of v to F ⊃ E to be any one inducing w on E , and we willaccordingly view the local reciprocity maps rec w : E × w = F × v → Gal( F v /F v ) ab = Gal( F v /E w ) ab . Let U = U ◦ F e U = Q v U v ⊂ B ∞× be an open compact subgroup and φ ∈ S ( B × A × ) satisfying Assumption 6.2.2 for integers r = ( r v ) v | p . Fix throughout this subsection a prime v | p .theBy Lemma 3.1.1, the generating series e Z ( φ ∞ ) is invariant under K ( ̟ r ′ ) v for some r ′ > . Wecompute the action of the operator U v, ∗ on it. DANIEL DISEGNI
Lemma 9.1.1 . —
For each a ∈ A × , v | p , the a t h reduced q -expansion coefficient of U v, ∗ e Z ( φ ∞ ) equals Z ( ̟ − v ) e Z a̟ v ( φ ∞ ) , where the ( a̟ v ) t h q -expansion coefficient of e Z ( φ ) is given in (5.2.2) .Proof . — We have U v, ∗ φ v ( x, u ) = | ̟ v | X j ∈ O F,v /̟ v r ( (cid:0) ̟ v j (cid:1) ) φ v ( x, u )= | ̟ v | φ v ( ̟ v x, ̟ − v u ) under our assumptions on φ v .Plugging this in the definition of e Z and performing the change of variables x ′ = ̟ v x we obtainthe result.We wish to give a more explicit expression for e Z a̟ sv ( φ ∞ )[1] U = c U p | a̟ s | X x ∈ U \ B ∞× /U φ ∞ ( x, a̟ sv q ( x ) − )[ x ] U (9.1.1)as s varies. Let Ξ( ̟ r v v ) = (cid:26)(cid:18) a bc d (cid:19) ∈ M ( O F,v ) | a, d ∈ ̟ r v v O F,v (cid:27) U ◦ F,v . Then Ξ( ̟ r v v ) ⊂ B v is the image of the support of φ v under the natural projection B v × F × v → B v ;and for each x v ∈ U v \ B × v /U v , some x v ∈ B v ∞ , × such that x v x v contributes to the sum (9.1.1)exists if and only if x v belongs to Ξ( ̟ r v v ) a̟ svv := { x v ∈ Ξ( ̟ r v v ) , q ( x v ) ∈ a̟ s v v (1 + ̟ r O F,v ) } . Lemma 9.1.2 . —
Let U v , φ v be as in Assumption 6.2.2, and let a ∈ F × v with s v = v ( a ) ≥ r v .Then the quotient sets U v \ Ξ( ̟ r v v ) a /U v , Ξ( ̟ r v v ) a /U v and U v \ Ξ( ̟ r v v ) a are in bijection, and foreach of them the set of elements x v ( b v , a ) := (cid:18) b v b − v (1 − a ) 1 (cid:19) ∈ M ( F v ) = B v , b v ∈ ( O F,v /̟ r v + s v v ) × is a complete set of representatives.Proof . — We drop the subscripts v . By acting on the right with diagonal elements belonging to U ,we can bring any element x ∈ Ξ( ̟ r ) a to one of the form x ( b, a ′ ) with b ∈ O × F , a ′ ∈ a (1 + ̟ r O F ) .The right action of an element γ ∈ U sends an element x ( b, a ) to one of the same form x ( b ′ , a ′ ) ifand only if γ = λ̟ r − bµ̟ r − a − λ̟ r b ̟ r µ ! for some λ, µ ∈ O F ; in this case we have b ′ = b − aµ̟ r − a , a ′ = a (cid:18) − ̟ r ( λ + µ ) − ̟ r λµa − a̟ r (cid:19) . The situation when considering the left action of U is analogous (as can be seen by the symmetry b ↔ b − (1 − u̟ s ) ). The lemma follows.Henceforth we will just write x v ( b v ) for x v ( b v , a ) unless there is risk of confusion. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES Lemma 9.1.3 . —
1. Let x = x v x v ( b v ) , c v ∈ O F,v = O E w ) . The action of the Galois element rec E (1 + c v ̟ r v v ) rec(1 + c v ̟ r v v )[ x ] U = [ x v x v ( b v (1 + c v ̟ r v v ))] U . (9.1.2)
2. For each a ∈ F × v with v ( a ) ≥ r , we have Ξ( ̟ r v v ) a /U v = a b ∈ ( O F /̟ rv ) × rec E ((1 + ̟ r v O F,v / ̟ s v O F,v )) x v ( b ) U v , where b is any lift of b to O F,v /̟ r v + s v .Proof . — Both assertions follow from the explicit description of the Galois action on CM points:we have rec E (1 + c v ̟ r v v )[ x ] U = (cid:20)(cid:18) c v ̟ rv (cid:19) x (cid:21) U , and a calculation establishes part 1. In view of Lemma 9.1.2, part 2 is then a restatement of theobvious identity ( O F v /̟ r + sv ) × = ̟ r O F,v ̟ s O F,v ( O F v /̟ rv ) × . Norm relation for the generating series . — Let e Z va ( φ v ) := c U p X x v ∈ B v ∞× /U φ v ∞ ( x v , aq ( x v ) − ) Z ( x v ) U . Then we have e Z a̟ s ( φ )[1] U = | ̟ v | − r v v X x v e Z va ( φ v )[ x v ] U where the sum runs over x v ∈ Ξ( ̟ r v v ) v ( a )+ v ( d )+ s v /U v .For s ≥ r v , let H s ⊂ E ab be the extension of E with norm group U ◦ F U vT (1 + ̟ sw O E,w ) , where U T = U ∩ E × A ∞ . Let H ∞ = ∪ s ≥ r v H s . If r v is sufficiently large, for all s ≥ r v the extension H s /H r v is totally ramified at w , and we have Gal( H s /H r ) ∼ = Gal( H s v ,w /H r v ,w ) ∼ = (1 + ̟ r v v O F,v ) / (1 + ̟ sv O F,v ) . (9.1.3)For convenience we set s ′ := v ( a ) + s , H ′ s := H r v + s ′ = H r v + v ( a )+ s for any s ∈ { , , . . . , ∞} , and we denote by Tr s , and similarly later N s , the trace (respectivelynorm) with respect to the field extension H ′ s /H ′ . Proposition 9.1.4 . —
Fix any a ∈ A ∞ , × . With the notation just defined, we have e Z a̟ s ( φ )[1] U = X i ∈ I X b ∈ ( O F,v /̟ r ) × Tr s c i [ x vi x v ( b, a̟ s )] U where the finite indexing set I , the constants c i ∈ Q and the cosets x vi U v are independent of s .Moreover there exists an integer d = 0 independent of a such that c i ∈ d − Z for all i .Proof . — By Lemma 9.1.3 we can write e Z a̟ s ( φ )[1] U = | ̟ v | − r v v X b ∈ ( O F /̟ rvv ) × e Z va ( φ v ) [Gal( H ′ s,w /H ′ ,w ) · x v ( b, a̟ s )] U = | ̟ v | − r v v X b ∈ ( O F /̟ rvv ) × Tr s ( e Z va ( φ v )[ x v ( b, a̟ s )] U ) DANIEL DISEGNI using (9.1.3), as by construction the correspondence e Z va ( φ v ) U is defined over H ′ . We obtain theresult by writing e Z va ( φ v ) U = P i ∈ I c i Z ( x vi ) U .Finally, the existence of d follows from the fact that φ ∞ is a Schwartz function. The extension H ∞ ,w /E w . — After deShalit [17], given a non-archimedean local field K , we saythat an extension K ′ ⊂ K ab of K is a relative Lubin–Tate extension if there is a (necessarily unique)finite unramified extension K ⊂ K ◦ ⊂ K ′ such that K ′ ⊂ K ab is maximal for the property ofbeing totally ramified above K ◦ . By local class field theory, for any relative Lubin–Tate extension K ′ there exists a unique element ̟ ∈ K × with v K ( ̟ ) = [ K ◦ : K ] (where v K is the valuation of K ) such that K ′ ⊂ K ab is the subfield cut out by h ̟ i ⊂ K × via the reciprocity map of K . Wecall ̟ the pseudo-uniformiser associated with K ′′ . Lemma 9.1.5 . —
The field H ∞ ,w is the relative Lubin–Tate extension of E w associated with apseudo-uniformiser ̟ LT ∈ E × w which is algebraic over E and satisfies q w ( ̟ LT ) = 1 .Proof . — It is easy to verify that H ∞ ,w is a relative Lubin–Tate extension. We only need toidentify the corresponding pseudo-uniformiser ̟ LT . It suffices in fact to find an element θ ∈ E × satisfying q ( θ ) = 1 and lying in the kernel of rec E w : E × w → Gal( H ∞ ,w /E w ) , as then ̟ LT must bea root of θ hence also satisfies the required property.Let ̟ w ∈ E w be a uniformiser at w , and let d = [ E × A ∞ : E × U T ] , where U vT is as before, U T,w ⊂ O × E,w is arbitrary, and U T,w ∗ is identified with U ◦ F,v under F v ∼ = E w ∗ . Then we can find t ∈ E × , u ∈ U T such that ̟ dw = tu . We show that the image of θ := t/t in E × w lies in the kernelof rec E w : E × w → Gal( H ∞ ,w /E w ) . Letting ι w : E × w ֒ → E × A ∞ be the inclusion, we show equivalentlythat ι w ( θ ) is in the kernel of rec E : E × A ∞ → Gal( H ∞ ,w /E ) or concretely that i w ( t/t ) ∈ E × U vT U ◦ F .Now we have ι w ( t/t ) = t · u v ι w ( u w ∗ ) ι w ∗ ( u w ∗ ) . By construction u v ∈ U vT , and ι w ( u w ∗ ) ι w ∗ ( u w ∗ ) belongs to U ◦ F,v . This completes the proof.
Suppose still that the open compact U and the Schwartzfunction φ ∞ satisfy all of the assumptions of §6.1 together with Assumption 6.2.2. In this sub-section, we complete the proof of Theorem 8.3.2 by showing that the element e Z ( φ ∞ , χ )( p ) ∈ S isannihilated by ℓ ϕ p ,α . Let S be a finite set of non-archimedean places of F such that, for all v / ∈ S ,all the data are unramified, U v is maximal, and φ v is standard. Let K = K p K p be the level ofthe modular form e Z ( φ ∞ ) , and let T ι p ( σ ∨ ) ∈ H SL = H SM ⊗ M,ι p L be any element as in Proposition2.4.4.4. By that result, it suffices to prove that ℓ ϕ p ,α ( T ι p ( σ ∨ ) e Z ( φ ∞ , χ )( p )) = 0 . (9.2.1)We will in fact prove the following. Proposition 9.2.1 . —
For all v | p , the element T ι p ( σ ∨ ) e Z ( φ ∞ , χ )( v ) ∈ S ′ is v -critical in the senseof Definition 2.4.1. Recall that from §2.2 that the commutative ring H SM acts on the space of reduced q -expansions S ′ ( K p ) and its quotient S ′ S ( K p ) , so that the expression T ι p ( σ ∨ ) e Z ( φ ∞ , χ )( v ) makes sense. Propo-sition 9.2.1 implies that T ι p ( σ ∨ ) e Z ( φ ∞ , χ )( p ) is a p -critical element of S , hence it is annihilated by ℓ ϕ p ,α by Proposition 2.4.4.3, establishing (9.2.1).By Lemma 5.2.2, there is a Hecke correspondence T( σ ∨ ) U on X U (with coefficients in L ) suchthat T ι p ( σ ∨ ) q Z ∗ ( φ ∞ ) U = T ι p ( σ ∨ ) U ◦ Z ∗ ( φ ∞ ) U HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES as correspondences on X U . Then T ι p ( σ ∨ ) e Z ( φ ∞ , χ ) U ( v ) is an average of h T ι p ( σ ∨ ) U q e Z ∗ ( φ ∞ ) U [1] , t χ i w = h q e Z ∗ ( φ ∞ ) U [1] , T ι p ( σ ∨ ) t U t χ i w (9.2.2)for w | v .We study the class of T ι p ( σ ∨ ) t U t χ in H f ( E, Ind HE V p J U | G E ) , where H ⊂ E ab is any sufficientlylarge finite extension. Let L ′ denote Q p or any sufficiently large finite extension of Q p . As G F -representations, we have V p J U ⊗ Q p L ′ = M A ′ ,ι ′ : M A ′ ֒ → L ′ π UA ′ ⊗ M A ′ V p A ′ ι for some pairwise non-isogenous simple abelian varieties A ′ /F with End A ′ = M A ′ ; here V p A ′ ι := V p A ′ ⊗ M A ′ ,ι L ′ . More generally, let V := Ind HE V p J U | G H (9.2.3)for a finite extension H of E as above; then we have V ⊗ Q p L ′ = M A ′ ,ι ′ : M A ′ ֒ → L ′ ,χ ′ : Gal( H/E ) → L ′× π UA ′ ⊗ M A ′ V p A ′ ι ( χ ′ ) , where V p A ′ ι ( χ ′ ) := V p A ′ ⊗ M A ′ ,ι L ′ χ . Let V ′ := π UA ∨ ⊗ M V p A ∨ ι p ( χ − )) ⊂ V , where ι p : M ֒ → L ⊂ L ( χ ) is the usual embedding. Let V ′′ ⊂ V be its complement, V = V ′ ⊕ V ′′ . Proposition 9.2.2 . —
The class of T ι p ( σ ∨ ) t U t χ in H f ( E, V ) belongs to the subspace H f ( E, V ′ ) .Proof . — We may and do replace L ( χ ) by a sufficiently large finite extension L ′ . It is clear thatthe class of t χ belongs to H f ( E, V p J U ( χ − )) . Then it is enough to verify that T ι p ( σ ∨ ) t U annihilates L ( A ′ ,ι ) =( A ∨ ,ι p ) ( V p A ′ ι ) m A ′ . Equivalently, we show that for any ( A ′ , ι ) = ( A, ι p ) and any f ∈ π A ′ , f ∈ π A ′ , T alg ,ι ( f , f ) ◦ T ι p ( σ ∨ ) t U = 0 in Hom ( J U , J ∨ U ) .We have T alg ,ι ( f , f ) ◦ T ι p ( σ ∨ ) t U = T ι p ( σ ∨ ) t U ◦ T alg ,ι ( f , f ) = T ι p ( σ ∨ ) t U ◦ T alg ,ι ( θ ι ( ϕ ′ , φ ∞′ )) for some ϕ ′ ∈ σ ∞ A ′ and some rational Schwartz function φ ∞′ . By Theorem 5.4.1, this can berewritten as T ι p ( σ ∨ ) t U ◦ ( ιϕ ′ , e Z ∗ ( φ ∞′ )) σ ∞ A ′ = ( ιϕ ′ , T ι p ( σ ∨ ) t U ◦ e Z ∗ ( φ ∞′ )) σ ∞ A ′ using an obvious commutativity. Applying Lemma 5.2.2 again, we have T ι p ( σ ∨ ) t U ◦ e Z ∗ ( φ ∞′ ) = ι p T ( σ ) e Z ∗ ( φ ∞′ ) , where in view of (5.2.1), it is easy to see that we are justified in calling T ( σ ) the Hecke operatorcorresponding to T ( σ ∨ ) t U ; i.e., this Hecke operator acts as the idempotent projection onto σ K ⊂ M ( K, M ) for the appropriate level K . It is then clear that modular forms in the image of T ( σ ) are in the right kernel of ( , ) σ ∞ A ′ if σ A ′ = σ A ∨ or equivalently (as A ′ σ A ′ is injective) if A ′ = A ∨ .If A ′ = A ∨ , then the expression of interest is the image of ϕ ′ ⊗ T ( σ ) e Z ∗ ( φ ∞′ ) ∈ σ A ∨ ⊗ (Hom ( J U , J ∨ U ) ⊗ S ( M )) under the M -linear algebraic Petersson product and two projectionsapplied to the two factors, induced respectively from ι : M ⊗ L ′ → L ′ and ι p : M ⊗ L ′ → L ′ . If ι = ι p , their combination is zero. Proposition 9.2.3 . —
For each w | p and each a ∈ A S ∞ , × , the mixed extension E associatedwith the divisors e Z a ( φ ∞ ) U [1] and T ι p ( σ ∨ ) t χ is essentially crystalline at w . DANIEL DISEGNI
Proof . — By Proposition 4.3.1, it is equivalent to show that m w (cid:16) e Z a ( φ ∞ ) U [1] , T ι p ( σ ∨ ) t χ (cid:17) = 0 . Under Assumptions 6.1.2 and 6.1.3 (which are local at S ∪ S , hence unaffected by the action of T ι p ( σ ∨ ) ), this is proved in [60, Proposition 8.15, §8.5.1]. Proof of Proposition 9.2.1 . — For each w | v , by the discussion preceding (9.2.2) it suffices to showthat h q e Z ∗ ( φ ∞ ) U [1] , T ι p ( σ ∨ ) t U t χ i w is a v -critical element of S ′ , that is (by the definition and Lemma 9.1.1) that h e Z a̟ s ( φ ∞ ) U [1] , T ι p ( σ ∨ ) t U t χ i w ∈ q s − cF,v O L ( χ ) for a constant c independent of a and s .We may assume that w extends the place w of E fixed above.By Propositon 9.1.4, the divisor e Z a̟ s ( φ ∞ ) U [1] is a finite sum, with p -adically bounded coeffi-cients, of elements Tr s [ x j,s ] U , where Tr s denotes the trace map on divisors for the field extension H ′ s /H ′ , and [ x j,s ] U ∈ Div ( X U,H ′ s ) .Recall that h , i w = h , i ℓ w for a fixed ℓ w : F × w → L ( χ ) . (32) By (4.1.6), we have h Tr s [ x j,s ] U , T ι p ( σ ∨ ) t U t χ i ℓ w = h [ x j,s ] U , T ι p ( σ ∨ ) t U t χ i ℓ w ◦ N s,w (9.2.4)where we recall that N s = N s,w is the norm for H ′ s,w /H ′ ,w .Now take the field H of (9.2.3) to be H = H ′ s , and consider the G E w -representations V ′ ⊂ V = V s over L ( χ ) defined above Proposition 9.2.2. By our assumptions, V ′ satisfies the condition D pst ( V ′ ) ϕ =1 = 0 and the Panchishkin condition, with an exact sequence of G E w -representations → V + → V ′ → V − → . The representation V s has a natural G E -stable lattice, namely T s := Ind H ′ s E T p J U | G H ′ s . Let T ′ := T ∩ V ′ , T ′′ s := T s ∩ V ′′ , T ′ + := T ′ ∩ V + , T ′− = T ′ /T ′ + ; note that V ′ , T ′ , T ′± are independent of s (hence the notation).Let e E w /E w be a finite extension over which V p A (hence V ′ ) becomes semistable. We mayassume that the extension e E w ⊂ E ab is abelian and totally ramified (see [41, Proposition 12.11.5(iv), Proposition 12.5.10 and its proof]). For s ∈ { , , . . . , ∞} , let e H ′ s,w := e E w H ′ s,w be thecompositum.We can then apply Proposition 4.3.2, with the fields e H ′ s playing the role of the field denotedthere by ‘ F v ’; together with Proposition 9.2.3, it implies that (9.2.4) belongs to p − ( d + d ,s + d ) L ( χ ) ℓ w ◦ e N s (cid:16) O × H ′ s ,w ˆ ⊗ O L ( χ ) (cid:17) ⊂ p − ( d + d ,s + d + d ′ ) L ( χ ) q s − c F,v O L ( χ ) where d , d ,s , d are the integers of Proposition 4.3.2, (33) , e N s is the norm of e H ′ s / e H ′ , and d ′ accounts for the denominators coming from (4.1.7). The containment follows from the fact thatthe extension H ′ s /H ′ is totally ramified at w of degree q sF,v so that e H ′ s / e H ′ has ramification degreeat least q s − c F,v for some constant c .To complete the proof, we need to establish the boundedness of the integer sequence d ,s = length O L ( χ ) H ( e H ′ ,w , T ′′∗ s (1) ⊗ L ( χ ) / O L ( χ ) ) tors . (32) We only need to consider ℓ w on the field H ′ recalled just below. (33) Note that by construction there is an obvious direct sum decomposition T s = T ⊕ T s − for a complementarysubspace T s − ; so that the integer d is independent of s . HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES As H ( e H ′ ,w , T ′′∗ s (1) ⊗ L ( χ ) / O L ( χ ) ) tors ∼ = H ( e H ′ ,w , T ′′∗ s (1)) ⊂ H (cid:16) E w , T p J ∗ U (1) ⊗ O L ( χ ) [Gal( e H ′ s,w /E w )] (cid:17) , the boundedness follows from the next Lemma. Lemma 9.2.4 . —
Let Γ ′ LT := Gal( e H ′∞ ,w /E w ) . Then H ( E w , V p J ∗ U (1) ⊗ O L ( χ ) J Γ ′ LT K ) = 0 . Proof . — We use the results and notation of Lemma 9.1.5. Note first that we may safely replace L ( χ ) by a finite extension L ′ splitting E w / Q p . As V p J U = V p J ∗ U (1) is Hodge–Tate, we have H (cid:0) E w , V p J U ⊗ O L ( χ ) O L ( χ ) J Γ ′ LT K (cid:1) ⊂ M ψ H ( E w , V p J U ( ψ )) ( ψ − ) , where ψ runs through the Hodge–Tate characters of G E factoring through Γ ′ LT . Since the latter isa quotient of E × w dominating Γ LT = Gal( H ∞ ,w /E w ) ∼ = E × w / h ̟ LT i , we have Γ ′ LT ∼ = E × w / h ̟ e LT i forsome e ≥ . Then the condition that ψ factor through Γ ′ LT is equivalent to ψ ◦ rec E,w ( ̟ e LT ) = 1 for the psuedo-uniformiser ω LT ∈ E × w .By [51, Appendix III.A], a character ψ of G E,w is Hodge–Tate of some weight n ∈ Z [Hom ( E w , L ′ )] if and only if the maps E × w → L ′× given by ψ ◦ rec E,w and x x − n := Q τ ∈ Hom ( E w ,L ′ ) τ ( x ) − n ( τ ) coincide near ∈ E × w . By [15, Proposition B.4 (i)], ψ is crystalline if and only if those mapscoincide on O × E,w ; therefore we may write any Hodge–Tate character ψ as ψ = ψ ψ with ψ offinite order and ψ crystalline.Then, letting E w, the maximal unramified extension of Q p contained in E w and d = [ E w : E w, ] ,we can first write H ( E w , V p J U ( ψ )) = D crys ( V p J U ( ψ )) ϕ =1 ⊂ D crys ( V p J U ( ψ )) ϕ d =1 for the E w, -linear endomorphism ϕ d (where ϕ is the crystalline Frobenius), and then D crys ( V p J U ( ψ )) ϕ d =1 = D crys ( V p J U ( ψ )) ϕ d = λ − . where λ ∈ L ′ is the scalar giving the action of ϕ d on D crys ( ψ ) .By [38, Theorem 5.3], all eigenvalues of ϕ d on D crys ( V p J U ( ψ )) are Weil q E,w -numbers of strictlynegative weight. To conclude that D crys ( V p J U ( ψ )) ϕ d =1 = 0 it thus suffices to show that λ is analgebraic number of weight .By [15, Proposition B.4 (ii)], we have λ = ψ ◦ rec E,w ( ̟ w ) − · ̟ − nw where ̟ w ∈ E × w is any uniformiser and n are the Hodge–Tate weights. Writing ̟ m = u̟ LT for m = w ( ̟ LT ) and some u ∈ O × E,w , we have λ m = ψ ◦ rec E,w ( u̟ LT ) − · u − n ̟ − n LT . Now ψ ◦ rec E,w ( u ) = u − n by the crystalline condition , and ψ e ◦ rec E,w ( ̟ LT ) = 1 as ψ factorsthrough Γ ′ LT . Hence λ em = ̟ − en LT . By Lemma 9.1.5, ̟ LT is an algebraic number of weight ,hence so is λ .
10. Formulas in anticyclotomic families
In §§10.1-10.2, we prove Theorem C after filling in some details in its setup. It is largelya corollary of Theorem B (which we have proved for all but finitely many characters), once a DANIEL DISEGNI construction of the theta elements interpolating automorphic toric periods and Heegner points iscarried out. Finally, Theorem B for the missing characters will be recovered as a corollary ofTheorem C.In §10.3, we prove Theorem D.We invite the reader to go back to §1.4 for the setup and notation that we are going to use inthis section (except for the preliminary Lemma 10.1.1).
Let L and F be finite extensions of Q p , and denote by v thevaluation of F and by ̟ ∈ F a fixed uniformiser. Let π be a smooth representation of GL ( F ) on an L -vector space with central character ω and a stable O L -lattice π O L . Let E × ⊂ GL ( F ) be the diagonal torus. Assume that π is nearly ordinary in the sense of Definition 1.2.2 with unitcharacter α : F × → L × . Let f ◦ α ∈ π − { } be any nonzero element satisfying U ∗ v f ◦ α = α ( ̟ ) f ◦ α ,which is unique up to multiplication by L × . For r ≥ , let s r = (cid:18) ̟ r (cid:19) and f α,r := | ̟ | − r α ( ̟ ) − r s r f ◦ α . It is easy to check that f α,r is independent of the choice of ̟ , and it is invariant under V r = (cid:0) ̟ r O F,v (cid:1) V F , where V F := Ker ( ω ) ⊂ Z ( F ) . Lemma 10.1.1 . —
1. The collection f α,V r := f α,r , for r ≥ , defines an element f α = ( f α,V ) ∈ lim ←− V π V , where the inverse system runs over compact open subgroups V F ⊂ V ⊂ E × , and the transitionmaps π V ′ → π V are given by f
7→ − Z V/V ′ π ( t ) f dt.
2. Let π O L ⊂ π be a GL ( F ) -stable O L -lattice containing f ◦ α . The collection of elements ˜ f α,V r := | ̟ | r f α,r = α ( ̟ ) − r s r f ◦ α defines an element ˜ f α ∈ lim ←− V π V O L where the transition maps π V ′ → π V are given by f X t ∈ V/V ′ π ( t ) f. Proof . — We need to prove that − Z V r /V r +1 π ( t ) f α,r +1 dt = f α,r . (10.1.1)A set of representatives for V r /V r +1 is (cid:8)(cid:0) j̟ r (cid:1)(cid:9) j ∈ O F /̟ ; on the other hand recall that U ∗ v = P j ∈ O F /̟ v (cid:0) ̟ j (cid:1) . From the identity (cid:18) j̟ r (cid:19) (cid:18) ̟ r +1 (cid:19) = (cid:18) ̟ r (cid:19) (cid:18) ̟ j (cid:19) (cid:18) j̟ r (cid:19) we obtain − Z V r /V r +1 π ( t ) s r +1 f ◦ α = q − F,v s r U ∗ v f ◦ α = | ̟ | − α ( ̟ ) s r f ◦ α , as desired. The integrality statement of part 2 is clear as α ( ̟ ) ∈ O × L . HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES Let us restore the notation π + = π , π − = π ∨ , employing it in the current local setting. Then if π + is as in the previous Lemma, it is easy to check that π − is also nearly p ordinary. Explicitly,the element f + , ◦ α ( y ) = O F −{ } ( y ) | y | v α v ( y ) (10.1.2)in the L -rational subspace (34) of any Kirillov model of π + v satisfies U ∗ v f + , ◦ α = α v ( ̟ v ) f + , ◦ α , and theelement f − , ◦ α ( y ) = O F −{ } ( y ) | y | v ω − ( y ) α v ( y ) (10.1.3)in the L -rational subspace of any Kirillov model of π − v satisfies U v, ∗ f − , ◦ α = α v ( ̟ v ) f − , ◦ α .We can then construct an element f − α = ( f − α,V ) V = ( f − α,r ) r with the property of the previousLemma as f − α,r := | ̟ | − r α ( ̟ ) − r s ∗ r f − , ◦ ,α with s ∗ r = (cid:18) ̟ − r ̟ − r (cid:19) . Local toric periods . — Let us restore the subscripts v . Recall the universal Kirillov models K ( π ± v , ψ univ ,v ) of §2.3. Then the elements f ◦± α,v of (10.1.2), (10.1.3) yield, by the proof of theLemma, explicit elements f ± α,v ∈ lim ←− V K ( π ± v , ψ univ ,v ) V , (10.1.4)where the transition maps are given by averages.Recall the local toric period Q v ( f + v , f − v , χ v ) of (1.1.2), for a character χ v ∈ Y l . c .v , which wedefine on K ( π + v , ψ univ ,v ) ⊗ K ( π − v , ψ univ ,v ) using the canonical pairing of Lemma 2.3.2 on theuniversal Kirlillov models.By the previous discussion and the defining property of f ± α , for any character χ v ∈ Y l . c .v , theelement Q v ( f + α,v , f − α,v , χ v ) := lim V L (1 , η v ) L (1 , π v , ad) ζ F,v (2) L (1 / , π E,v ⊗ χ v ) v Z E × v /F × v χ v ( t )( π ( t ) f + α,v,V , f − α,v,V ) d ◦ t is well-defined and it belongs to M ( χ v ) ⊗ O Ψ v ( ω v ) . In fact, if M ( α v , χ v ) ⊂ L is a subfield containingthe values of α v , ω v , and χ v then Q v ( f + α,v , f − α,v , χ v ) belongs to o O Ψ v,M ( αv,χv ) ( ω v ) . Lemma 10.1.2 . —
With notation as above, we have Q v ( f + α,v , f − α,v ) = ζ F,v (2) − · Z ◦ v as sections of O Y v × Ψ v ( ω v ) ; here Z ◦ v is as in Theorem A.Proof . — It suffices to show that the result holds at any complex geometric point ( χ v , ψ v ) ∈ Y l . c .v × Ψ v ( C ) . Drop all subscripts v , and fix a sufficiently large integer r (depending on χ ).Recalling the pairing (2.3.2), we have by definition Q ( f + α , f − α , χ ) = L (1 , η ) L (1 , π, ad) ζ F (2) L (1 / , π E , χ ) Z E × /F × χ ( t )( π ( t ) f + α , f − α ) dt | d | / . We denote by E w (respectively E w ∗ ) the image of F under the map F → M ( F ) sending t ( t ) (respectively t ( t ) ), and by χ w (respectively χ w ∗ ) the restriction of χ to E × w (respectively E × w ∗ ).We can then compute that ζ F (1) L (1 / , π E , χ ) L (1 , η ) Q ( f + α , f − α , χ ) (34) Cf. §2.3. DANIEL DISEGNI equals | d | − Z F × Z F × | ̟ | − r α ( ̟ ) − r s r f + , ◦ α ( ty ) · | ̟ | − r α ( ̟ ) − r s ∗ r f − , ◦ α ( y ) χ w ( t ) d × y d × t = | d | − Z F × Z F × | ̟ | − r α ( ̟ ) − r ψ ( − ty ) | ty̟ r | α ( ty̟ r ) O F −{ } ( ty̟ r ) · | ̟ | − r α ( ̟ ) − r ψ ( y ) ω ( ̟ ) r | y̟ r | α ( y̟ r ) ω − ( y̟ r ) O F −{ } ( y̟ r ) χ w ( t ) d × y d × t. We now perform the change of variables t ′ = ty and observe that χ w ( t ) = χ w ( t ′ ) χ − w ( y ) = χ w ( t ′ ) ω ( y ) χ w ∗ ( y ) ; we conclude after simplification that the above expression equals | d | − Z v ( t ′ ) ≥− r ψ ( − t ′ ) | t ′ | α ( t ′ ) χ w ( t ′ ) d × t ′ Z v ( y ) ≥− r ψ ( − y ) | y | α ( y ) χ w ∗ ( y ) d × y. If r is sufficiently large, the domains of integration can be replaced by F × . The computation ofthe integrals is carried out in Lemma A.1.1. We obtain Q v ( f + α,v , f − α,v , χ v ) = L (1 , η v ) ζ F,v (1) L (1 / , π E,v , χ v ) ζ E,v (1) Y w | v Z w ( χ w ) = ζ F,v (2) − · Z ◦ v ( χ v ) . Here we prove Theorem C. We continuewith the notation of the previous subsection, and we suppose that π ± v is isomorphic to the localcomponent at v | p of the representation π ± of the Introduction. Let w , w ∗ be the two places of E above v , and fix an isomorphism B v ∼ = M ( F v ) such that the map E v ∼ = E w ⊕ E w ∗ → B v isidentified with the map F v ⊕ F v → M ( F v ) given by ( t , t ) (cid:0) t t (cid:1) .We go back to the global situation with the notation and assumption of §1.4. Choose a universalWhittaker (or Kirillov) functional for π + at p , that is, a B × p -equivariant map K + p : π + ⊗ O Ψ p (Ψ p ) → N v | p K ( π + , ψ univ ,v ) . By the natural dualities of π ± and the Kirillov models, it induces a B × p -equivariant map K + ∨ p : N v | p K ( π − , ψ univ ,v ) → π − ⊗ O Ψ p (Ψ p ) , whose inverse K − p is a universalKirillov functional for π − p . Letting π ± p, O Ψ p (Ψ p ) := N v | p K ( π ± , ψ univ ,v ) , we obtain a unique decom-position π ± ⊗ O Ψ p (Ψ p ) ∼ = π ± ,p O Ψ p (Ψ p ) ⊗ π p, O Ψ p (Ψ p ) . The decomposition arises from a decompositionof the natural M -rational subspaces π ± ∼ = π ± ,p ⊗ π ± p . (35) Theta elements . — For each f ± ,p ∈ π ± p ⊗ M ( α ) , let f ± ,pα := f ± p ⊗ f ± α,p ∈ π ± ,pM ( α ) ⊗ lim ←− V ⊂ O × E,p π ± ,Vp, O M ( α ) , where f ± α,p = ⊗ v | p f ± α,v with f ± α,v the elements (10.1.4).Fix a component Y ◦± ⊂ Y ± of type ε ∈ { +1 , − } as in §1.4. Then the elements Θ ± α ( f ± ,p ) := − Z E × \ E × A ∞ f p, ± α ( i ( t )) χ univ ( t ) dt ∈ O Y ± ( Y ± ) b , (10.2.1) P ± α ( f ± ,p ) := lim U T,p →{ } − Z E × \ E × A ∞ /U T κ ( f p, ± α (rec E ( t ) ι ξ ( P )) ⊗ χ ± univ ,U T ( t )) dt ∈ S p ( A E , χ ± univ , Y ± ) b , (10.2.2) (35) However the M -rational subspaces are not stable under the B ∞× -action. HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES or rather their restriction to Y ◦± ⊂ Y ± , satisfy the property of Theorem C.1. Here χ ± univ ,U T : Γ → O × ( Y ± ) U T is the convolution of χ ± univ with the finest U T -invariant approximation to a delta func-tion at ∈ Γ .We explain the boundedness in the case of P ± α ( f ± ,p ) . The rigid space Y ± is the genericfibre of an O L -formal scheme Y ± := Spf O L J Γ K / (( ω ± ( γ )[ γ ] − [1]) γ ∈ A ∞ , × ) . (36) (Similarly eachgeometric connected component Y ◦± has a formal model Y ◦± ⊂ Y ). This identifies O Y ± ( Y ± ) b = O Y ± ( Y ± ) ⊗ O L L . By Lemma 10.1.1.2 applied to the natural lattice π ± O L ⊂ π ± ⊗ L given by Hom (
J, A ± ) ⊗ End ( A ) O L ⊂ Hom ( J, A ± ) ⊗ L , after possibly replacing f ± ,p by a fixed multiplethe elements ˜ f p, ± α (T t ι ξ ( P )) belong to A ± ( E ab ) . Then each term in the sequence at the right-handside of (10.2.2) is a fixed multiple of X t ∈ E × \ E × A ∞ /U T κ ( ˜ f p, ± α (rec E ( t ) ι ξ ( P )) ⊗ χ ± univ ,U T ( t )) , (10.2.3)which belongs to the O L -module H f ( E, T p A ± ⊗ O Y ± ( Y ± ) U T ( χ ± univ ,U T )) . Hence some nonzeromultiple of P ± α ( f ± ,p ) belongs to the limit lim ←− U T H f ( E, T p A ± ⊗ O Y ± ( Y ± ) U T ( χ ± univ ,U T )) , whosetensor product with L is indeed S p ( A E , χ ± univ , Y ± ) b . Local toric periods away from p . — Given the chosen decomposition γ ± : π ± ⊗ O Ψ p (Ψ p ) ∼ = π ± ,p O Ψ p (Ψ p ) ⊗ π ± p, O Ψ p (Ψ p ) , let ( , ) p be the unique pairing on π + ,p O Ψ p ⊗ π − ,p O Ψ p which makes γ + ⊗ γ − into an isometryfor the natural pairings on π ± and π ± p, O Ψ p (Ψ p ) . (37) Then for each χ = χ p χ p ∈ Y ◦ l . c . , the toricperiod Q p of (1.4.5) is defined. By Lemma 5.1.1, Theorem C.2 then follows from Proposition 3.6.1.It is also proved in slightly different language in [35, Lemma 4.6 (ii)]. Formulas . — We prove the anticyclotomic formulas of Theorem C, and at the same time completethe proof of Theorem B for the characters who do not satisfy Assumption 6.2.4.
Lemma 10.2.1 . —
Let L be a non-archimedean local field with ring of integers O L , n ≥ , and let D n be the rigid analytic polydisc over L in n variables, that is the generic fibre of Spf O L J X , . . . , X n K .Let Σ n ⊂ D n ( L ) be the set of points of the form x = ( ζ − , . . . , ζ n − with each ζ i a root of unityof p -power order. Let f ∈ O ( D n ) b = O L J X , . . . , X n K ⊗ L be such that f ( x ) = 0 for all but finitely may x ∈ Σ n . Then f = 0 .Proof . — By induction of n , the case n = 1 being well-known [3]; we will abbreviate X =( X , . . . , X n ) and X ′ = ( X , . . . , X n − ) . Up to multiplying f by a suitable nonzero polyno-mial we may assume that f ∈ O L J X K and that it vanishes on all of Σ n . We may write, withmultiindex notation, f = P J ⊂ N n a J X J (where N = { , , , . . . , } ) with each a J ∈ O L . Let f j ( X ′ ) := P J ′ ⊂ N n − a J ′ j ( X ′ ) J ′ ∈ O L J X ′ K , then f ( X ) = ∞ X j =0 f j ( X ′ ) X jn . By assumption, f ( X ′ ) = f ( X ′ , vanishes on all of Σ n − , hence by the induction hypothesis f = 0 and X n | f . By induction on j , repeatedly replacing f by X − n f , we find that each f j = 0 ,hence f = 0 . (36) The quotienting ideal is finitely generated as the image of A ∞ , × in Γ is a finitely generated Z p -submodule. (37) Recall that the natural pairing on the factor at p comes from its description as a Kirillov model. DANIEL DISEGNI
Lemma 10.2.2 . —
Let Y ◦ ⊂ Y be a connected component of type ε = − , and let Y ◦′ ⊂ Y ′ be the connected component containing Y ◦ . The p -adic L -function L p,α ( σ E ) | Y ◦′ is a section of I Y ⊂ O Y ′ .Proof . — By the interpolation property and the functional equation, L p,α ( σ E ) vanishes on Y ◦ ∩ Y l . c ., an ( L ) . We conclude by applying Lemma 10.2.1, noting that after base-change to a finiteextension of L , there is an isomorphism Y → ` i ∈ I D ( i )[ F : Q ] to a finite disjoint union of rigidpolydiscs, taking Y l . c ., an ( L ) to ` i ∈ I Σ ( i )[ F : Q ] . Proposition 10.2.3 . —
The following are equivalent:1. Theorem B is true for all f , f and all locally constant characters χ ∈ Y l . c .L ;2. Theorem B is true for all f , f and all but finitely many locally constant characters χ ∈ Y l . c .L ;3. Theorem C.4 is true for all f + ,p and f − ,p .Proof . — It is clear that 1 implies 2. That 2 implies 3 follows from Lemma 10.2.1 applied to thedifference of the two sides of the desired equality, together with the interpolation properties andthe evaluation of the local toric integrals in Lemma 10.1.2. Finally, the multiplicity one resulttogether with Lemma 10.1.2 shows that 3 implies 1.Since we have already shown at the end of §8.3 that Theorem B is true for all but finitely manyfinite order characters, this completes the proof of Theorem B in general and proves Theorem C.4.Finally, the anticyclotomic Waldspurger formula of Theorem C.3 follows from the Waldspurgerformula at finite order characters (1.4.1) by the argument in the proof of Proposition 10.2.3. Theorem D follows immediately from combin-ing the first and second parts of the following Proposition. We abbreviate S ± p := S p ( A ± E , χ ± univ , Y ◦ ) b ,and remark that, under the assumption ω = of Theorem D, we have A = A + = A − and π = π + = π − . Proposition 10.3.1 . —
Under the assumptions and notation of Theorem D, the following hold.1. Let H ⊂ S + p ⊗ S − ,ι p be the saturated Λ -submodule generated by the Heegner points P + α ( f p ) ⊗ P − α ( f p ) for f p ∈ π p .The Λ -modules S ± p are generically of rank one, and moreover H is free of rank over Λ ,generated by an explicit element P + α ⊗ P − ,ια .2. We have the divisibility of Λ -ideals char Λ e H f ( E, V p A ⊗ Λ( χ univ )) tors | char Λ (cid:0) S + p ⊗ Λ S − ,ι p / H (cid:1) .
3. Letting (cid:10) (cid:11) denote the height pairing (1.4.4) , we have (cid:10) P + α ⊗ P − ,ια (cid:11) = c E · d F L p,α ( σ E ) | Y ◦ . Proof . — We define e Q v ( f v , χ v ) := Q v ( f v , f v , χ v ) for f v ∈ π v , and similarly e Q ( f, χ ) := Q v e Q ( f v , χ v ) if f = ⊗ v f v ∈ π . By [56, Lemme 13] (possibly applied to twists ( π v ⊗ µ − ‘ v , χ v · ( µ v ◦ q w )) for some char-acter µ v of F × v ), the spaces H( π v , χ v ) = Hom E × v ( π v ⊗ χ v , L ( χ v )) are nonzero if and only if e Q v ( · , χ v ) is nonzero on π v . We also define e Q v ( f v ) := Q v ( f v , f v ) and e Q ( f p ) := ζ F,p (2) − Q v ∤ p e Q ( f v ) ∈ Λ for f p = ⊗ f v ∈ π p . If the local conditions (1.1.1) are satisfied, as we assume, then the spaces H( π v , χ v ) are nonzero for all χ ∈ Y ◦ , and e Q is not identically zero on π p .We will invoke the results of Fouquet in [22] after comparing our setup with his. Let U r = U p Q v | p U r,v ⊂ B ∞ , × be such that U r,v = K ( ̟ r v v ) for v | p , r v ≥ . Let eH ét ( X U r ,F , O L (1)) HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES be the image of H ét ( X U r ,F , O L (1)) under the product e of the projectors e v := lim n U n ! v . Let J π ⊂ H sph B ∞× be the annihilator of π viewed as a module over the spherical Hecke algebra H sph B ∞× for B ∞× , and let eH ét ( X U p ,F , O L (1))[ π ] := eH ét ( X U r ,F , O L (1)) / J π , a Galois-module which is independent of r ≥ . The operators U v act invertibly on eH ( X U r ,F , O L (1)) ,and in fact by α v ( ̟ v ) on eH ét ( X U p ,F , O L (1))[ π ] . Let f p ∈ π p be such that e Q ( f p ) = 0 . Denoteby κ the Abel–Jacobi functor and by f ◦ α := f p ⊗ f ◦ α,p , with f ◦ α,p the product of the elements f ◦ α,v of(10.1.2) for v | p . Then, up to a fixed nonzero multiple, the class P + α ( f p ) is the image under κ ( f ◦ α ) of the limit of the compatible sequence P α,r := U − rp X t ∈ E × \ E × A ∞ /U T κ (rec E ( t )T s r ι ξ ( P )) ⊗ χ ± univ ,U T ( t )) of integral elements of H f ( E, eH ét ( X U p ,F , O L (1)) ⊗ Λ U T ( χ univ ,U T )) . Fouquet takes as input acertain compatible sequence ( z ( c p , S )) S ([22, Definitions 4.11, 4.14]) of classes in the latter spaceto construct, via suitable local modifications according to the method of Kolyvagin, an Eulersystem ( ibid. §5). Noting that the local modifications occur at well-chosen, good primes ℓ of E ,his construction can equally well be applied to the sequence ( P α,r ) r in place of ( z ( c p , S )) S . ThisEuler system can then be projected via κ ( f α ) to yield an Euler system for V p A ⊗ Λ . Under thecondition that the first element P + α ( f p ) ∈ S + p (corresponding to z f, ∞ in [22]) of the projectedEuler system is non-torsion, it is proved in [22, Theorem B (ii)] that S ± p have generic rank over Λ . By the main result of [1], generalising [16], the family of points P + α ( f p ) ∈ S + p is indeed not Λ -torsion provided e Q ( f p ) = 0 (i.e. f v is a “local test vector” for all v ∤ p ). We conclude as desiredthat S ± p have generic rank over Λ and that the same is true of the submodule H .We now proceed to complete the proof of part 1 by showing that the ‘Heegner submodule’ H isin fact free of rank and constructing a ‘canonical’ generator. First we note that by [22, Theorem6.1], each special fibre H | χ (for arbitrary χ ∈ Y ◦ ) has dimension either (we will soon exclude thiscase) or over L ( χ ) . Let { P + α ( f pi ) ⊗ P − α ( f pi ) ι : i ∈ I } be finitely many sections of H . By [60],for each χ ∈ Y ◦ , l . c . := Y ◦ ∩ Y l . c ., an , the specialisation P + α ( f pi ) ⊗ P − α ( f pi ) ι ( χ ) is nonzero if andonly if e Q ( f pi )( χ ) = 0 , and moreover the images of the specialisations at χ of the global sections of H Y j ∈ I,j = i e Q ( f pj ) · P + α ( f pi ) ⊗ P − α ( f pi ) ι (10.3.1)under the Néron–Tate height pairing (after choosing any embedding L ( χ ) ֒ → C ) coincide. As H | χ has dimension at most , the Néron–Tate height pairing on H χ ⊗ C is an isomorphism onto itsimage in C , and we deduce that that the elements (10.3.1) coincide over Y ◦ , l . c . . Since the latterset is dense in Spec Λ by Lemma 10.2.1, they coincide everywhere and glue to a global section ( P + α ⊗ P − ,ια ) ′ of H over Y ◦ . (38) Similarly to what claimed in the proof of Theorem 3.7.1, there exists a finite set { f pi : i ∈ I } ⊂ π p such that the open sets U f pi := { e Q ( f pi ) = 0 } ⊂ Y ◦ cover Y ◦ . Then the section P + α ⊗ P − ,ια := Y i ∈ I e Q ( f pi ) − · ( P + α ⊗ P − ,ια ) ′ is nowhere vanishing and a generator of H . It is independent of choices since for any f p ∈ π p itcoincides with e Q ( f p ) − · P + α ( f p ) ⊗ P − α ( f p ) ι over U f p . This completes the proof of Part 1. (38) We remark that a similar argument, in conjunction with the previous observation that e Q = 0 on π p if and onlyif Q = 0 on π p ⊗ π p , shows that H equals the a priori larger saturated submodule H ′ ⊂ S + p ⊗ Λ S − ,ι p generatedby the P + α ( f + ,p ) ⊗ P − α ( f − ,p ) ι for possibly different f ± ,p ∈ π . DANIEL DISEGNI
Part 2 is [22, Theorem B (ii)] with H replaced by its submodule generated by a non-torsionelement z f, ∞ ⊗ z f, ∞ ∈ H ; as noted above we can replace this element by any of the elements P + α ( f p ) ⊗ P − α ( f p ) ι , and by a glueing argument with P + α ⊗ P − ,ια .Part 3 is an immediate consequence of Theorem C.4. Appendix ALocal integralsA.1. Basic integral. —
All the integrals computed in the appendix will ultimately reduce tothe following.
Lemma A.1.1 . —
Let F v be a non-archimedean local field, and let E w /F v be an extension ofdegree f e ≤ , with f the inertia degree and e the ramification degree. Let q w be the relative normand D ∈ O F,v be a generator of the relative discriminant.Let L be a field of characteristic zero, let α v : F × v → L × and χ ′ : E × w → L × be multiplicativecharacters, ψ v : F v → L × be an additive character of level , and ψ E,w = ψ v ◦ Tr E w /F v . Define Z w ( χ ′ , ψ v ) := Z E × w α ◦ q ( t ) χ ′ ( t ) ψ E w ( t ) dt | D | / | d | f/ v . where dt is the restriction of the standard measure on E w .Then we have: Z w ( χ ′ , ψ v ) = α v ( ̟ v ) − v ( D ) χ ′ w ( ̟ w ) − v ( D ) − α v ( ̟ v ) − f χ ′ w ( ̟ w ) − − α v ( ̟ v ) f χ ′ w ( ̟ w ) q − fF,v if χ ′ w · α v ◦ q is unramified, τ ( χ ′ w · α v ◦ q, ψ E w ) if χ ′ w · α v ◦ q is ramified.Here for any character e χ ′ w of conductor f , τ ( e χ ′ , ψ E w ) = Z w ( t )= − w ( f ) e χ w ( t ) ψ E,w ( t ) dt | D | / | d | f/ v . with n = − w ( f ( χ ′ w )) − w ( d E,w ) .Proof . — If χ ′ is unramified, only the subset { w ( t ) ≥ − − w ( d ) − v ( D ) } ⊂ E × w contributes to theintegral, and we have Z w ( χ ′ , ψ ) = α − v ( D ) χ ′ w ( ̟ w ) − ev ( d ) − v ( D ) ζ E (1) − − α f χ ′ ( ̟ ) − q fF,v α − f χ ′ ( ̟ ) − ! = α − v ( D ) χ ′ w ( ̟ w ) − ev ( d ) − v ( D ) − α − f χ ′ ( ̟ ) − q − fF,v − α f χ ′ ( ̟ ) . If χ ′ is ramified of conductor f = f ( χ ′ ) then only the annulus w ( t ) = − w ( f ) − w ( d ) − v ( D ) contributes,and we get Z w ( χ ′ , ψ ) = α − fw ( f ) − fv ( D ) v τ ( χ ′ , ψ E ) . A.2. Interpolation factor. —
We compute the integral giving the interpolation factor for the p -adic L -function.The following Iwahori decomposition can be proved similarly to [30, Lemma A.1]. Lemma A.2.1 . —
For a local field F with uniformiser ̟ , and for any r ≥ , the double quotient N ( F ) A ( F ) Z ( F ) \ GL ( F ) /K ( ̟ r ) HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES admits the set of representatives (cid:18) (cid:19) , (cid:18) c ( i ) ̟ r − i (cid:19) , ≤ i ≤ r, c ( i ) ∈ ( O F /̟ i ) × . Note that we may also replace the first representative by (cid:0) ̟ r (cid:1) ∈ K ( ̟ r ) . Proposition A.2.2 . —
Let χ ′ ∈ Y ′ l . c .M ( α ) ( C ) and let ι : M ( α ) ֒ → C be the induced embedding.Let v | p , and let φ v be either as in Assumption 6.2.1 for some sufficiently small U T,v ⊂ O × E,v ,or as in Assumption 6.2.2. Then for any sufficiently large integer r , the normalised integral R ♮r,v ( W v , φ v , χ ′ v ) of (3.5.2) equals R ♮r,v ( W v , φ v , χ ′ vι ) = Z ◦ v ( χ ′ v ) := ζ F,v (2) L (1 , η v ) L (1 / , σ ιE,v , χ ′ v ) Y w | v Z w ( χ ′ w ) , with Z w ( χ ′ w ) as in Lemma A.1.1.Proof . — We omit the subscripts v and the embedding ι in the calculations which follow. Bydefinition, we need to show that the integral R ◦ r,v of Proposition 3.5.1 equals R ◦ r,v = R ◦ r ( W, φ, χ ′ ) = | D | / | d | L (1 , η ) Y w | v Z w ( χ ′ w ) . Note that the assertion in the case of Assumption 6.2.2 is implied by the assertion in the case ofAssumption 6.2.1 by (6.2.1), so we will place ourselves in the latter situation.By the decomposition of Lemma A.2.1, and observing that δ χ F ,r vanishes on K − K ( ̟ r ) , wehave R ◦ r,v = α ( ̟ ) − r Z F × W − ( (cid:0) y (cid:1) ) δ χ F ,r ( (cid:0) y (cid:1) ) Z T ( F ) χ ′ ( t ) Z P ( ̟ r ) \ K ( ̟ r ) | y | r ( kw − r ) φ ( yt − , y − q ( t )) dk d × t d × y | y | where P ( ̟ r ) = P ∩ K ( ̟ r ) (recall that P = N ZA ). Here we have preferred to denote by d × t thestandard Haar measure on T ( F v ) = E × v ; later dt will denote the additive measure on E v .Changing variables k ′ = w r kw − r , we observe that by Lemma 3.1.1 the group K ( ̟ r ) actstrivially for sufficiently large r . Then we can plug in W − ( (cid:0) y (cid:1) ) = O F −{ } ( y ) | y | α ( y ) and r ( w − r ) φ ( x, u ) = | ̟ − r | ψ E,U ( ux ) O V ( x ) δ q ( U ) ( ̟ r u ) , where ψ E,U = r (vol( U ) − U ) ψ E for the extension of r to functions on K (so that ψ E,U is thefinest U -invariant approximation to ψ E ). We obtain R ◦ r,v = α ( ̟ ) − r | d | / ζ F,v (1) − Z O F −{ } | y | α ( y ) Z T ( F ) χ ′ ( t ) ψ E,U ( t ) δ q ( U ) ( ̟ r y − q ( t )) dt d × y, where | d | / ζ F,v (1) − appears as vol( P ( ̟ r ) \ K ( ̟ r )) | ̟ | − r . We get R ◦ r,v = | d | ζ F,v (1) − Z v ( q ( t )) ≥− r | q ( t ) | α ( q ( t )) χ ′ ( t ) ψ E ( t ) d × t. If r is sufficiently large, the domain of integration can be replaced with all of T ( F ) . Switching tothe additive measure, and using the isomorphism E × v = E × w × E × w ∗ in the split case, the integral DANIEL DISEGNI equals R ◦ r,v = | d | L (1 , η ) Z E × v α ( q ( t )) χ ′ ( t ) ψ E ( t ) dt = | D | / | d | L (1 , η ) Y w | v Z w ( χ ′ w ) as desired. A.3. Toric period. —
We compare the normalised toric periods with the interpolation factor.
Proposition A.3.1 . —
Suppose that v | p splits in E . Then for any finite order character χ ∈ Y l . c . we have Q v ( θ v ( W v , α − r v v w − r,v φ v ) , χ v ) = L (1 , η v ) − · Z ◦ v ( α v , χ v ) , for any φ v is as in Proposition A.2.2 and any sufficiently large r v ≥ . For consistency with the proof of Proposition A.2.2, in the proof we will denote by d × t the Haarmeasure on T ( F ) denoted by dt in the rest of the paper. Proof . — By the definitions and Proposition A.2.2, it suffices to show that for any χ ∈ Y l . c . ( C ) we have | d | / v Q ♯v ( θ v ( W v , α − r v v w − r,v φ v ) , χ v ) = L (1 , η v ) − · R ◦ r,v ( W v , φ v , χ v ) where Q ♯v is the toric integral of (5.1.5).By the Shimizu lifting (Lemma 5.1.1) and Lemma A.2.1 we can write Q ♯v := | d | / v Q v ( θ v ( W v , α − r v v w − r,v φ v ) , χ v ) = Q ♯ (0 , v + r X i =1 X c ∈ ( O F /̟ i ) × Q ♯ ( i,c ) v where for each ( i, c ) , omitting the subscripts v , Q ♯ ( i,c ) v := α ( ̟ ) − r Z F × W − ( (cid:0) y (cid:1) n − ( c̟ r − i )) Z T ( F ) χ ′ ( t ) Z P ( ̟ r ) \ K ( ̟ r ) | y | r ( n − ( c̟ r − i ) kw − r ) φ ( yt − , y − q ( t )) dk d × t d × y | y | . Note that Q ♯ (0 , v = R ◦ v , where R ◦ v is as in the previous Proposition, since n − ( ̟ r ) ∈ K ( ̟ r ) .We will compute the other terms.We have (cid:18) c̟ r − i (cid:19) w − r = w − r (cid:18) − c̟ − i (cid:19) , and when x = ( x , x ) with x = 0 : r ( n − ( c̟ r − i ) w − r ) φ ( x, u ) = | ̟ − r | Z E ψ E ( ux ξ ) ψ ( − uc̟ r − i q ( ξ )) δ U ( ξ ) δ q ( U ) ( ̟ r u ) d ξ Z O V ψ ( − uc̟ r − i q ( ξ )) δ q ( U ) ( ̟ r u ) d ξ = | ̟ | i − r ψ E,U ( ux ) ψ q ( U ) ( − c̟ − i ) δ q ( U ) ( ̟ r u ) . Plugging this in, we obtain Q ♯ ( i,c ) v = | ̟ | i α ( ̟ ) − r | d | / ζ F,v (1) − Z F × W ( (cid:0) y (cid:1) n − ( c̟ r − i )) Z T ( F ) χ ( t ) · ψ E,U ( t ) ψ q ( U ) ( − c̟ − i ) δ q ( U ) ( ̟ r y − q ( t )) d × t d × y, We have already noted that Q ♯ (0 , v = R ◦ v . For i = 1 , if r is sufficiently large then W is still invariantunder K ( ̟ r − ) ; then P c Q ♯ (1 ,c ) v equals C · R ◦ v with C = | ̟ | P c ∈ ( O /̟ ) × ψ q ( U ) ( − c̟ − ) = −| ̟ | . HE p -ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES Finally, we claim that for each i ≥ , P c Q ♯ ( i,c ) v = 0 . Indeed let q ( U ) = 1 + ̟ n O F,v . If i ≥ n + 1 , then ψ q ( U ) ( − c̟ − i ) = 0 ; if i ≤ n , then if r is sufficiently large W is still invariantunder K ( ̟ r − i ) ⊂ K ( ̟ r − n ) , and summing the only terms depending on c produces a factor P c ∈ ( O /̟ i ) × ψ ( c̟ − i ) = 0 .Summing up, we have Q ♯v = Q ♯ (0 , v + X c ∈ ( O F /̟ v ) × Q ♯ (1 ,c ) v = (1 − | ̟ | ) R ◦ r,v = L (1 , η v ) − · R ◦ r,v as desired. Question A.3.2 . — A comparison between Propositions 10.1.2 and A.3.1 suggests that the iden-tity lim r →∞ L (1 , η v ) · θ v ( W v , α − rv w − r φ v ) , χ v ) = ζ F,v (2) · f + α,v ⊗ f − α,v . might hold in lim ←− V ( π + v ) V ⊗ lim ←− V ( π − v ) V (with notation as in Lemma 10.1.1). Is this the case? References [1] Esther Aflalo and Jan Nekovář,
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