aa r X i v : . [ m a t h . N T ] A p r Λ -ADIC FAMILIES OF JACOBI FORMS MATTEO LONGO, MARC-HUBERT NICOLE
Abstract.
We show that Hida’s families of p -adic elliptic modular forms generalize to p -adic families of Jacobi forms. We also construct p -adic versions of theta lifts from ellipticmodular forms to Jacobi forms. Our results extend to Jacobi forms previous works by Hidaand Stevens on the related case of half-integral weight modular forms. Introduction
The theory of p -adic families of ordinary cusp forms was initiated by Hida ([Hid86a],[Hid86b], [Hid88]), starting with the group GL(2). Over the last 30 years, its scope ex-tended greatly, encompassing groups associated to Shimura data of Hodge type, and beyond.Historically though, some of the first extensions of the theory were outside the realm of re-ductive groups, for example the metaplectic group governing half-integral weights modularforms, worked out by Hida himself in [Hid95] by d´evissage to his original theory. Anotherimportant theme of p -adic interpolation has been to establish variants in families of variousfunctorial liftings between spaces of modular forms. For half-integral weight modular forms,the p -adic variant of the theta lift due originally to Shintani was studied by Stevens in [Ste94].In this paper, we wish to substitute, in the non-reductive examples mentioned above, half-integral weight modular forms with Jacobi forms, and verify that the corresponding objectsand liftings vary equally well in ordinary families, thus extending some results of Guerzhoy[Gue00a].In this paper we therefore have two aims: (1) Develop a theory of p -adic families of ordinaryJacobi forms, following the approach of Hida [Hid95]; (2) Construct, following [Ste94], anexample of theta lifts from p -adic families of ordinary elliptic modular forms to p -adic familiesof ordinary Jacobi forms which interpolates the Shimura-Shintani correspondence for classicalforms.Before explaining our results in more details, we stress that one of the motivations to developsuch a theory, especially the p -adic theta correspondence, lies with companion paper [LN19]which investigates an analogue for p -adic families of the Gross-Kohnen-Zagier theorem (GKZfor short), in which Heegner points are substituted with p -adic families of Heegner points calledBig Heegner points. It seems that a satisfactory, although still conjectural, generalisation ofthe GKZ Theorem for Big Heegner points involves the p -adic theta correspondence that weconstruct in this paper. In Section 4 we describe in a more precise way the relation betweenthis paper and [LN19].In the first part of the paper, we develop concisely the general framework for p -adic familiesof Jacobi forms, relying on unpublished work [Canb] applied only in the much simpler andmuch better studied case of dimension one. This part can be seen as an extension to Jacobiforms of the analogous theory developed by Hida in [Hid95] for half-integral weight modularforms. The main result of this first part is Corollary 2.12, which is an analogue of thecelebrated Hida Control Theorem ([Hid88, Theorem II]) in the context of p -adic familiesof ordinary Jacobi cusp forms. Corollary 2.12 shows that the Λ-module of p -adic familiesof Jacobi forms specializes to vector spaces of classical Jacobi forms when it is cut out byarithmetic primes. We now state a simplified form of the Control Theorem for Jacobi forms.Fix an integer N ≥ p ∤ N . Let Λ = O [[1 + p Z p ]] be the Iwasawa algebra of 1+ p Z p with coefficients in the valuation ring O of a fixed finite extension of Q p . We considera normal integral domain I which is finite over Λ. Put X ( I ) = Spec( I )( C p ). A point κ ∈ X ( I )is said to be arithmetic if its restriction to 1 + p Z p ⊆ I is of the form γ ψ ( γ ) γ k − for a finiteorder character ψ of 1 + p Z p (called the wild character of κ ) and an integer k ≥ κ ). We introduce the metaplectic cover ˜ I = I ⊗ Λ ˜Λ, where ˜Λ is the Λ-algebra whichis equal to Λ as Λ-module, but the Λ-algebra structure is given by ( λ, x ) λ x for λ ∈ Λ and x ∈ ˜Λ. We say that a point in ˜ X ( I ) = Spec(˜ I )( C p ) is arithmetic if it maps to an arithmeticpoint of X ( I ) via the canonical projection map ˜ X ( I ) → X ( I ). A I -adic Jacobi form of index N is a formal power series in S = X r r ≤ nN c n,r q n ζ r ∈ ˜ I [[ q, ζ ]] , where c n,r ∈ ˜ I , such that, for each arithmetic point κ ∈ ˜ X ( I ), the specialization of S ( κ )obtained by evaluating the coefficients c n,r at κ , is the ( q, ζ )-expansion of a classical Jacobiform, with coefficients in C , after fixing an algebraic isomorphism C ≃ C p which we choosefrom now on. We call classical specialisations those classical Jacobi forms arising as speciali-sations of a I -adic form. Restricting to those I -adic forms whose classical specialisations areordinary cuspidal forms, we define the submodule of I -adic ordinary cuspidal forms, denoted J cusp , ord N ( I ). Let J cusp , ord k,N ( p s , χ ) be the C -vector space of Jacobi cuspidal ordinary forms ofweight k , index N , level p s and character χ ; we refer to § I is primitive, meaning that classical specialisations in a dense subset ofthe weight space are newforms (see Def. 2.9 (4)). Theorem 1.1.
Let I be primitive. The Λ = O [[1 + p Z p ]] -module J cusp , ord N ( I ) is finitely gener-ated, and for each arithmetic point κ ∈ ˜ X ( I ) we have J cusp , ord N ( I ) / p κ J ord , cusp N ( I ) ≃ J cusp , ord k,N ( p s , χ ) where p κ is the kernel of κ , and the integers k, s and the character χ depend on κ only. In the second part of the paper, we set up a p -adic theta correspondence relating ordinary p -adic families of elliptic cuspforms with ordinary p -adic families of Jacobi forms. Recall that theclassical theta correspondence associates, under certain arithmetic conditions, a Jacobi form toan elliptic modular form; see § p -adic version of this theta correspondence. The main result of this part isTheorem 3.6, which shows the existence of a p -adic family of Jacobi forms interpolatingclassical theta lifts of the classical forms in a given p -adic family of elliptic ordinary cuspforms.This p -adic version of the theta correspondence has been developed for p -adic half-integralweight modular forms by Stevens [Ste94] in the elliptic case, as recalled above, and the authors[LN13] in the quaternionic case. To state our main result in a more precise form, recall thatfor an elliptic modular form f ∈ S k (Γ ( M ) , χ ) of weight k , level Γ ( M ) for some integer M ,and character χ , we have a Shintani lift f ( χ )( D ,r ) ( f )to the space of Jacobi forms of prescribed weight, index, character and level depending on f and the choice of a fundamental discriminant D which is a square modulo N , a square root r of D modulo N , and a square root χ of the character χ ; the precise definition of the map S ( χ )( D ,r ) is recalled in § f ∞ = P n ≥ a n q n ∈ I [[ q ]] of ordinary elliptic cuspforms; recall that for each arithmetic point κ ∈ X ( I ) the specialization f κ = P n ≥ a n ( κ ) q n isan elliptic eigenform of level Γ ( N p s ), character χ and weight k , where the integers k, s and -ADIC FAMILIES OF JACOBI FORMS 3 the character χ depend on κ only, and p s is the maximum between 1 and the p -power of theconductor of χ . Theorem 1.2.
There exists a p -adic family of Jacobi forms S such that for each ˜ κ ∈ ˜ X ( I ) lying over an arithmetic point κ ∈ X ( I ) of character χ with χ a non-trivial character modulo p , we have S (˜ κ ) = λ ( κ ) · S ( χ ) D ,r ( f κ ) where λ ( κ ) ∈ C p is a p -adic period which is non-zero on an open subset of X ( I ) . A more general form of Theorem 1.2 is presented in Theorem 3.6 and Theorem 3.7 whichconsider generic arithmetic points κ of non-trivial character χ and trivial character, respec-tively. These results show that theta lifts can be interpolated in families. We also discuss aresult, see Theorem 3.10, where we consider similar questions for forms which arise in Hidafamilies as ordinary p -stabilisations of forms of level N prime to p . Acknowledgments.
Part of this work was done during visits of M.-H.N. at the MathematicsDepartment of the University of Padova whose congenial hospitality he is grateful for, and alsoduring a visit of M.L. in Montr´eal supported by the grant of the CRM-Simons professorshipheld by M.-H.N. in 2017-2018 at the Centre de recherches math´ematiques (C.R.M., Montr´eal).2. Λ -adic families of Jacobi forms
In this section, we give an account of some aspects of Hida theory for families of Jacobiforms relying on the well-known close relationship between Jacobi forms and modular formsof half-integral weight. Explicit examples of Λ-adic families of Jacobi forms are provided bytheta lifts of Λ-adic families of classical modular forms, and we provide them later in thenext section. It should be no surprise that Hida theory extends to the setting of Jacobi forms:Wiles’s convolution trick with the classical Λ-adic Eisenstein series readily shows the existenceof a Λ-adic family of Jacobi forms specializing to a given p -ordinary Jacobi form, see Section2.7 for details.We rely on two main tools to study p -adic families of Jacobi forms:(1) first, Candelori’s unpublished results in the preprint [Canb] applied to classical mod-ular curves to relate algebraically Jacobi forms and vector-valued half-integral weightmodular forms. While [Canb] treats the higher-dimensional case using the languageof stacks, all we need can be formulated plainly and with completeness in terms ofmodular curves without relying on his more sophisticated techniques.(2) second, Hida’s treatment for scalar-valued half-integral weight modular forms [Hid95].Recall that Hida developped in [Hid95] his eponymous theory for scalar-valued half-integral weight modular forms, while by Eichler-Zagier’s theorem and its generaliza-tions, Jacobi forms are more naturally related to vector-valued half-integral weightmodular forms. Since the Jacobi group is not reductive, the geometric approachworking well to develop Hida theory for Shimura varieties does not seem to be di-rectly applicable (but see [Kra95] for groundwork on the arithmetic theory of Jacobiforms), hence our heavy reliance on Hida’s paper [Hid95] for content and overall strat-egy. Note that for Jacobi forms, the usual role of the tame level is played by the index m = N , using the notations below.2.1. Jacobi forms.
We review the basic definition of Jacobi forms. The Jacobi group is thesemi-direct product SL ( Z ) ⋉ Z where the law is defined by:( γ, Z )( γ ′ , Z ′ ) = ( γγ ′ , Zγ ′ + Z ′ ) . MATTEO LONGO, MARC-HUBERT NICOLE
The Jacobi group acts on the
H × C . Let Γ J := Γ ⋉ Z , Γ a congruence subgroup of SL ( Z ).Let Z = ( λ, µ ) ∈ Z , and γ = (cid:0) a bc d (cid:1) ∈ Γ . For ( γ, Z ) ∈ Γ J and ( τ, z ) ∈ H × C , let( γ, Z ) · ( τ, z ) = (cid:16) aτ + bcτ + d , z + λτ + µcτ + d (cid:17) . This defines a right action of Γ J on H × C . Further, the Jacobi group acts on complexfunctions on H × C via the slash operator φ | k,m for integers k, m ∈ N i.e., ( γ,
0) acts as( cτ + d ) − k e (cid:16) − cmz cτ + d (cid:17) φ (cid:16) aτ + bcτ + d , zcτ + d (cid:17) , and (1 , ( λ, µ )) acts as e ( λ mτ + 2 λmz ) φ ( τ, z + λτ + µ ) , where e ( − ) represents the exponential e πi − . Definition 2.1.
Let k, m ∈ N . A Jacobi form of weight k and index m for the congruencesubgroup Γ is a function: φ : H × C −→ C which satisfies the following condition:(1) it satisfies φ | k,m ( γ, Z ) = φ for all ( γ, Z ) ∈ Γ J .(2) it is holomorphic on H × C ;(3) it is holomorphic at the cusps.Here, we refer to [EZ85, Def. p.9] for a precise explanation of the behaviour at the cuspsin terms of the Fourier expansion. Further, a Jacobi cusp form is defined as Jacobi formvanishing at all cusps in a suitable sense. The C -vector space of Jacobi (resp. cusp) forms ofweight k and index m for a congruence subgroup Γ is denoted by J k,m (Γ) (resp. J cusp k,m (Γ)) byslight abuse of notation.2.2. The Eichler-Zagier theorem.
We describe in this section an account of results by[Canb] which allow to derive geometrically a higher level version of the classical Eichler-Zagier theorem. We state results in [Canb] in the 1-dimensional case of modular curves;the reader is advised to keep a copy of [Canb] when reading this subsection to compare oursetting with the more general results developed in loc cit. , see also [Can14] for backgroundand more details. The strategy, aimed to introduce p -adic families of Jacobi forms, is toadd level structures of increasing p -power levels to obtain an analogue of [Canb, Thm.4.2.1].After introducing analogous level structures on the C -vector space of vector-valued modularforms, and passing to global sections over C , we recover a generalization of the Eichler-Zagiertheorem in higher level (see also [Boy15]). A more general version of these results couldalso be obtained by introducing more general level structure; however, adding p -power levelstructure is enough for our applications to p -adic families of Jacobi forms. As mentionedabove we provide some of the details to accomplish this algebraically in the technically muchsimpler 1-dimensional case, see [Canb, Exa. 2.6.10, Sect. 2.8] for the simplifications occuring,especially the “accidental isomorphism” of [Canb, Eq. (2.8)] which does not extend to higherdegree. We recall a few definitions from [Canb] but we specialize them readily to modularcurves, cf. [Ram06]. We shall formulate directly the results in term of compact modularcurves X (4 N ) and X ( p s ), obtained as compactification of the open modular curves Y (4 N )and Y ( p s ) associated to the congruence subgroups Γ (4 N ) and Γ ( p s ), for an integer s ≥ -ADIC FAMILIES OF JACOBI FORMS 5 optimal, canonical results (also available through more elementary, classical techniques alone)once specializing to C .Let N ∈ N be an integer and p ∤ N , p ≥ A , (Θ), introducedin [Canb, Def. 2.6.6], is (isomorphic to) the classical open moduli stack Y (4 N ) of pairs( E, H ) of elliptic curves E equipped with a subgroup scheme H of E [4 N ] locally isomorphicto Z / N Z . The theta group is defined asΓ(1 ,
2) := (cid:8)(cid:0) a bc d (cid:1) ∈ SL ( Z ) | ab, cd ≡ (cid:9) . Following [Canb, Def. 2.6.1], we define the stack X , as classifying elliptic curves togetherwith a symmetric, relatively ample, normalized invertible sheaf L of degree 1. The stack X , maps to the moduli stack of elliptic curves, and it is smooth over Z [1 / X , has twoconnected components: we therefore have a decomposition X , = X +1 , ⊔ X − , . More precisely,as it is pointed out in [Canb, p. 16], over C we have that: X + , an1 , = Γ(1 , \H , X − , an1 , = SL ( Z ) \H . Here the superscript an denotes the associated complex analytic space. We denote by themore customary notation Y (Γ(1 , X +1 , the so-called “even characteristic” connectedcomponent of the stack defined over Z [1 / Y (4 N ) −→ X , defined in greater generality using the moduliinterpretations, see [Canb, Defining equation (2.9)], and it factors through X +1 , by [Canb, Sec.2.7]. Over modular curves, we can define it in the following concrete way: Definition 2.2. ([Canb, (2.9) and Lemma 2.8.6]) We denote byDes C : Y (4 N ) C −→ Y (Γ(1 , C , the analytification of the descent map induced by the homomorphism between the fundamentalgroups: Γ (4 N ) −→ Γ(1 , , Des (cid:18) a bc d (cid:19) = (cid:18) a bNc/ N d (cid:19) . We now introduce the automorphic sheaves giving rise to Jacobi forms, resp. half-integralweight modular forms.On the Jacobi side, the basic invertible sheaf is L N , and for simplicity we describe itusing complex algebraic geometry i.e., using the complex uniformization (see [Kra91] forthe treatment of general level structures and [Kra95] for an approach relying on arithmeticgeometry including the treatment of compactifications). Consider the elliptic modular surfacegiven by the quotient S Γ ( p s ) := H × C / (Γ ( p s ) ⋉ Z ) , where ( m, n ) ∈ Z acts by ( τ, z ) ( τ, z + m + nτ ), and γ acts by γ ( τ, z ) = (cid:16) aτ + bcτ + d , zcτ + d (cid:17) , γ = (cid:18) a bc d (cid:19) ∈ Γ ( p s ) . Define L N := O S (2 N e S ) ⊗ (cid:16) Ω S /Y ( p s ) (cid:17) N , where e S is the identity section. Let π : S → Y ( p s )denote the projection morphism, and define J , N := π ∗ L N . Let E → Y ( p s ) denote theuniversal elliptic curve over Y ( p s ). Definition 2.1.
Let N ≥ k ≥ N and weight k with respect to Γ ( p s ) is a global section of J , N ⊗ ω ⊗ k E /Y ( p s ) . MATTEO LONGO, MARC-HUBERT NICOLE
On half-integral weight side, the construction involves a few more ingredients. To tacklehalf-integral weight modular forms geometrically, the standard trick is to fix a square root ofthe Hodge bundle ω . This is the strategy exploited by Hida in [Hid95]. Candelori’s so-calledmetaplectic stacky formalism is a generalisation of this idea. The canonical bundle over X , of weight 1/2 forms denoted ω / is very closely connected to the square root of the Hodgebundle ω (see [Canb, Cor. 3.3.3]), hence the notation. Its definition is as follows: the invertiblesheaf ω / is defined on [Canb, p.22, line 3] as the canonical square root of M Θ ⊗ ω induced bythe canonical isomorphism of [Cana, Thm. 5.0.1], where M Θ is the theta multiplier bundleof [Canb, Def. 3.2.2]. That is, the choice of the square root of ω is made compatibly with thechoice of the square root of M Θ so that their tensor product is canonical.More concretely, the bundle Des ∗ ω / can be seen as an invertible sheaf of modular formsof half-integral weight 1 / Y (4 N ), prompting the following: Definition 2.2. ([Canb, Def. 3.4.1]) Let k ≥ k/ ∗ ω k/ .We now introduce vector-valued modular forms on the modular curve Y ( p s ). The newingredient is the so-called Weil bundle. Over C , this is the local system corresponding to theWeil representation, and for modular curves, it is defined as follows. Let H (2 N ) := Z / N Z ,and K (2 N ) := Z / N Z × Z / N Z . The Heisenberg group of type 2 N is defined as G (2 N ) := G m × K (2 N )with group structure given by ( λ , x , y )( λ , x , y ) = ( λ λ h x , y i N , x + x , y + y ), where h− , −i is the standard symplectic pairing of type 2 N defined in [Canb, Eq. (2.4)]. Let S be anyscheme. The sheaf V (2 N ) is the free O S -module of functions f : H (2 N ) → O S , with canonicalbasis given by delta functions. The Schr¨odinger representation of G (2 N ) is defined in [Canb,Def. 4.1.2] as the module V (2 N ) given by functions f : H (2 N ) → O S , with G (2 N )-actiongiven by: ρ ( λ, x, y ) f ( y ′ ) := λ h x, y ′ i f ( y ′ + y ) . The sheaf V (2 N ) is the Schr¨odinger representation of weight one and rank 2 N . We can nowdefine the Weil bundle W , N as the locally free sheaf V (2 N ) ∨ ⊗ Des ∗ M − / ⊗ ω − / , see[Canb, Eq. (4.3)]. The key point of [Canb, Sec. 4.3] is that the Weil bundle is defined overthe metaplectic stack (even though neither V (2 N ) ∨ nor Des ∗ M − / is), i.e., it is defined over Y ( p s ) after fixing a square root of the Hodge bundle ω over Y ( p s ). Definition 2.3.
Let k ≥ W , N -vector valued modular form of weight k/ W , N ⊗ ω k +1 / . Remark . The above definition of vector-valued modular forms of weight k/ Theorem 2.3.
There is a canonical isomorphism J , N ∼ = −→ W , N , as locally free sheaves of rank N over Y ( p s ) C .Proof. The key point in the above construction is that both sheaves are generally defined overthe metaplectic stack over the modular elliptic curve Y ( p s ), that is, over Y ( p s ) itself once asquare root of the Hodge bundle ω E /Y ( p s ) is fixed. The isomorphism [Canb, (4.3)] states that V (2 n ) ⊗ Des ∗ M − / ∼ = J , N ⊗ ω / , -ADIC FAMILIES OF JACOBI FORMS 7 and it is canonical thanks to [Cana, Thm 5.0.1]. (cid:3) Remark . [Canb, Thm. 4.2.1, (4.3)] refine an old result of Mumford, see [Canb, Sect. 4.4]and [MB90] for details. We work over C as the most precise result is established up to ± X ( p s ) (cf. [MB90, Sect. 2.5] for the computation at the cusps with Tate curves), we obtainthe following slight extension to level Γ ( p s ) of the classical Eichler-Zagier isomorphism, see[EZ85, Thm.5.1], cf. [Canb, Cor.4.5.1] and see [Boy15, §
3, Thm. 3.5] for further generalizationover C : Corollary 2.5.
There is a canonical isomorphism between the space J k,N (Γ ( p s )) of Jacobiforms of index N , weight k and level Γ ( p s ) and the space of W ∨ , N -vector-valued modularforms of weight k − / , denoted V k − / , N (Γ ( p s )) .Proof. We tensor the isomorphism V (2 N ) ⊗ Des ∗ M − / ∼ = J , N ⊗ ω / with ω ⊗ k E /Y ( p s ) on both sides and obtain the canonical isomorphism: V (2 N ) ⊗ Des ∗ M − / ⊗ ω ⊗ k − / E /Y ( p s ) ∼ = J , N ⊗ ω ⊗ k E /Y ( p s ) . The sheaves and the isomorphism extend to the compactification X ( p s ). Taking globalsections yields the canonical isomorphism between the two spaces of modular forms. (cid:3) The vector-valued modular forms in V k − / , N (Γ ( p s )) above generalize the 2 N -tuples ofmodular forms as in [EZ85, p. 59] with respect to SL ( Z ).2.3. Density of classical Jacobi forms.
As the theory of Jacobi forms is not fully ge-ometrized in comparison with the theory of classical modular forms, we face similar difficul-ties as for half-integral weight modular forms where the underlying non-reductive group isthe metaplectic group. Our strategy to prove the density theorem for classical Jacobi formswithin p -adic Jacobi forms is to use the natural isomorphism `a la Eichler-Zagier between thespace of Jacobi forms and the space of vector-valued half-integral weight modular forms. Moreprecisely, we show that the completion of the space of p -power level classical Jacobi forms doesnot depend on the weight, and therefore any such classical subspace is dense in the space of p -adic Jacobi forms, see Thm 2.6 for the precise statement.We set up the necessary notation for the density theorem. Let J k,N ( A ) := [ s J k,N (Γ ( p s ); A ) , J cusp k,N ( A ) := [ s J cusp k,N (Γ ( p s ); A ) , for A any subalgebra of C , and where J k,N (Γ ( p s ); A ) denotes the space of Jacobi forms ofweight k , index N and level Γ ( p s ) with Fourier-Jacobi coefficients in A .Let W ( F p ) be the ring of Witt vectors with residue field F p , and K the quotient field of W ( F p ). We fix once and for all a complex (resp. p -adic) embedding of ¯ Q into C (resp. ¯ Q p ).We may thus take A := W ( F p ) ∩ Q and J k,N ( W ( F p )) = J k,N ( A ) ⊗ A W ( F p ), J cusp k,N ( W ( F p )) = J cusp k,N ( A ) ⊗ A W ( F p ). We write b J k,N ( W ( F p )) for the p -adic completion of J k,N ( W ( F p )), andsimilarly for cusp forms. The following theorem proves that the p -adic completion is indepen-dent of the weight k . Theorem 2.6 (Density theorem) . For k ≥ , we have an isomorphism preserving the Fourier-Jacobi coefficients: b J cusp k,N ( W ( F p )) ∼ = b J cusp k +1 ,N ( W ( F p )) . MATTEO LONGO, MARC-HUBERT NICOLE
Proof.
The first step is to use Corollary 2.5 to transfer the problem to vector-valued modularforms. Note that the construction of vector-valued modular forms or taking the p -adic comple-tion are operations that commute with each other, since the corresponding vector spaces havefinite dimension. Using [Hid95, Thm.1, p.145], we already know the result for scalar-valuedhalf-integral weight modular forms, and this implies that the space of classical vector-valuedmodular forms is dense in the space of p -adic vector-valued modular forms by using the pro-jections to the scalar-valued components. The result for Jacobi forms follows. (cid:3) Ordinary forms.
Let N ≥ p ∤ N , p > s ≥ χ : ( Z /p s Z ) × → ¯ Q × p a Dirichlet character. Denote U J ( p ) the Hecke operator at p acting on the space of Jacobi forms J cusp k,N ( p s , χ ) of index N , character χ , with respect toΓ ( p s ), see [MRV93] and [Ibu12] for details. By [MRV93, §
3, Eq. (5), p.165], if the ( n, r )-thFourier-Jacobi coefficient of a U J ( p )-eigenform F is c ( n, r ), then the ( n, r )-th Fourier-Jacobicoefficient of F | U J ( p ) is c ( p n, pr ). Definition 2.7.
We say that a Jacobi eigenform F ∈ J cusp k,N ( p s , χ ) is p -ordinary if its U J ( p )-eigenvalue is a p -adic unit.We denote J cusp , ord k,N ( p s , χ ) the C -vector space of ordinary Jacobi cuspforms. We startwith recalling the notion of newforms in the Jacobi case introduced in [MR00]. Denote S new2 k − (Γ ( N p s ) , χ ) the space of newforms of weight 2 k −
2, level Γ ( N p s ) and character χ .As in [MR00, Sect. 5.1], let f , . . . , f t be an orthonormal basis of S new2 k − (Γ ( N p s ) , χ ) ofnormalized Hecke eigenforms, and write f i = P n ≥ a n ( f i ) q n . Let J cusp , new k,N ( p s , χ, f i ) be thesubspace of J cusp k,N ( p s , χ ) consisting of forms F such that F | T J ( ℓ ) = a ℓ ( f i ) · F for all primes ℓ ∤ N p . Define J cusp , new k,N ( p s , χ ) = t M i =1 J cusp , new k,N ( p s , χ, f i ) . Theorem 2.8 (Weak control theorem for Jacobi newforms) . Let k ≥ . The number oflinearly independent p -ordinary Jacobi cuspidal newforms in J cusp , new k,N ( p s , χ ) is bounded aboveindependently of the weight k .Proof. Using [MR00, Thm. 5.2], we embed the space of Jacobi newforms which are p -ordinaryinjectively into the space of half-integral weight newforms via the generalized Eichler-Zagiermap denoted by Z N (or rather Z m ) in [MR00]. Since the map Z N preserves p -ordinarity,the image is in the space of p -ordinary half-integral weight newforms. To conclude, we apply[Hid95, Prop.3] which relies heavily on results of Waldspurger via the Shimura correspondence. (cid:3) Hida families for GL . We set up the notation for Hida families of modular forms. Let N ≥ p > p, N ) = 1. Let f ∞ ( κ ) = X m ≥ a n ( κ ) q n ∈ R [[ q ]]be a primitive Hida family, where R is a primitive branch of the ordinary Big Hida Heckealgebra acting of forms of tame level Γ ( N ) with coefficients in the ring of integers O of afinite extension F of Q p ; for details and definitions, see [How07, § § R is an integral complete local noetherian domain, which is finitely generated over theIwasawa algebra Λ = O [[1 + p Z p ]], and such that for each arithmetic weight κ in X arith ( R ) ⊆ X ( R ) = Hom cont O -alg ( R , ¯ Q p ) -ADIC FAMILIES OF JACOBI FORMS 9 of R , the formal Fourier expansion f κ = P n ≥ a n ( κ ) q n , where a n ( κ ) := κ ( a n ), is an ordinary p -stabilized newform of level N . As a general notation, for a ∈ R and κ ∈ X ( R ) we write a ( κ ) for κ ( a ). We also recall that a homomorphism κ ∈ X ( R ) is said to be arithmetic if itsrestriction to 1 + p Z p has the form γ ψ ( γ ) γ k − for a finite order character ψ (called thewild character of κ ) and an integer k ≥ κ ). In this case the weight of f κ is k , its level is Γ s = Γ ( N ) ∩ Γ ( p s ) where s ≥ ψ and 1, and the coefficients a n ( κ ) belong to the finite extension R κ := R p κ / p κ R p κ of Q p ,where p κ := ker( κ ) and R p κ is the localization of R at κ . We also introduce the notation O f κ for the valuation ring of R κ . We call ( χ, k ) the signature of the arithmetic character κ , see[Ste94, Def. 1.2.2].2.6. Hida families of Jacobi forms.
We recall the notation used in [Hid95]. Let as aboveΛ be the complete group algebra of 1 + p Z p with coefficients in O , which is non-canonicallyisomorphic to the one-variable power series ring in a variable X with coefficients in O viathe map u X , after fixing a generator u ∈ p Z p . Fix an algebraic closure L of thequotient field L of Λ. For each normal integral domain I in L finite over Λ (for example, onecan take I = R with R as in § X ( I ) be weight space i.e., the space of C p -valued pointsof Spec( I ). We say that a weight κ ∈ X ( I ) is arithmetic if its restriction to Λ is, in the senseof the previous section. Definition 2.5.
The metaplectic cover of I is the I -algebra ˜ I = I ⊗ Λ ˜Λ, where ˜Λ = Λ asΛ-module, but the Λ-algebra structure is given by ( λ, x ) λ x for λ ∈ Λ and x ∈ ˜Λ.Put ˜ X ( I ) = Spec(˜ I )( C p ). We have a canonical surjective map π : ˜ X ( I ) ։ X ( I ), whichmakes ˜ X ( I ) a 2-fold cover of X ( I ). Definition 2.6.
We say that a point ˜ κ ∈ ˜ X ( I ) is arithmetic if π (˜ κ ) is arithmetic. Remark . Our examples arising from theta liftings in the next section are naturally ex-pressed in terms of the metaplectic cover e X ( I ) obtained from twisting the Λ-algebra structureby σ ( t ) = t on 1 + p Z p , because of the choices related to lifting data, see [Ste94, § § Definition 2.9. (1) A Λ-adic Jacobi form of index N over I , or simply a I -adic modularform, is a formal power series in ˜ I [[ q, ζ ]]: F ∞ (˜ κ ) = X r ≤ nN c n,r (˜ κ ) q n ζ r , such that the specializations F ˜ κ := F ∞ (˜ κ ) at almost all arithmetic primes κ ∈ ˜ X ( I )are images under the fixed embedding Q ֒ → Q p of Fourier expansions of classicalJacobi forms of index N , weight k , level p s , for some integer s , and character χ κ ofconductor dividing N p s , which are eigenforms for all Hecke operators. We call classicalspecialisations those classical Jacobi forms which arise as specialisations at arithmeticprimes. We finally write J N ( I ) for the I -module of I -adic modular forms.(2) The I -module of p -ordinary I -adic Jacobi forms is the I -submodule of J N ( I ) consistingof I -adic forms whose all classical specialisations are ordinary.(3) The I -module J ord , cusp N ( I ) of ordinary I -adic Jacobi cusp forms is defined as the I -submodule of the I -module J ord N ( I ) consisting of formal power series whose classicalspecialisations are cuspforms.(4) We say that I is primitive if for any F ∈ J ord , cusp N ( I ) there is a Zariski open set U ∈ ˜ X ( I )such that F ( κ ) is a newform for every κ ∈ U .Similarly, the module of cusp forms is defined by adding the condition of being cuspidal.We denote J ord , cusp N ( I ) the I -module of ordinary cuspidal forms. Remark . Examples of primitive Λ-algebras I arise naturally when one considers, as wewill do in the next section, Theta lifts of primitive branches of Hida families. Theorem 2.11.
Let I be primitive. The module of Λ -adic ordinary cusp forms J ord , cusp N ( I ) isfree of finite rank over the Iwasawa algebra Λ .Proof. As in [Hid95], we follow the argument of Wiles. The main observation is that wecan pick an arithmetic weight such that all the specializations of a maximal finite set oflinearly independent elements in the space of p -ordinary forms are newforms, and then applyTheorem 2.8. We give the details by paraphrasing Hida’s account in [Hid95, Prop.4, p. 159].Let J cusp , ord N ( I ) ⊗ K be the K -vector space of p -ordinary Jacobi modular forms of index N ,where K is the fraction field of I . Let F , . . . , F m be a finite set of linearly independentelements in J cusp , ord N ( I ) over I , with Fourier coefficients denoted by c F j ( n, r ). Then we can findpairs of integers ( n i , r i ), i = 1 , . . . , m so that D = det( c F j ( n i , r i )) = 0 , ≤ i, j ≤ m. By the density theorem and the fact that I is primitive we may choose an arithmetic weight κ so that for all i = 1 , . . . , m , F i ( κ ) ∈ J new , ord κ,N ( p s ( κ ) , χ ( κ )) and D ( κ ) = 0 . Since D ( κ ) = det( c F j ( n i , r i )( κ )) = 0, the specializations F i ( κ ) are also linearly independent.That is, m ≤ dim J new , ord κ,N ( p s ( κ ) , χ ( κ )) , which is bounded independently of κ by Theorem 2.8. Thus, there is a maximal set of linearlyindependent elements in J ord , cusp N ( I ). The rest of the proof is identical as in [Hid95]. (cid:3) Corollary 2.12.
Suppose I is primitive. For all arithmetic points in X ( I ) such that κ issufficiently large, we have: J cusp , ord N ( I ) / p κ J cusp , ord N ( I ) ∼ = J cusp , ord k ( s ) ,N ( p s ( κ ) , χ ) . Proof.
The surjectivity is guaranteed by Wiles’s trick whose details we recall in the nextSubsection, and the rest of the argument follows as in [Hid95, p.161]. (cid:3)
Remark . It might be possible to obtain a more general version of the above theorem andcorollary in which the primitivity assumption is suppressed. This could be accomplished bymeans of [MR00, Theorem 5.16], which describes, under certain conditions, the decompositionof Jacobi forms into a direct sum of newforms and oldforms, and using the analogue of Theorem2.8 for all C -vector spaces of Jacobi forms appearing in this decomposition. However, to obtainan analogue of Theorem 2.8 one needs to understand the kernel of the Eichler-Zagier map,which is not injective on oldforms, and bound it effectively, at least under some arithmeticconditions. In our applications in this paper and in [LN19], we only use primitive branches.2.7. Wiles’s trick for Jacobi forms.
Wiles’s trick consist in convoluting a classical modularform f with the Λ-adic Eisenstein series to show the existence of a Hida family specializing to f . Viewing the Eisenstein family as indexing a family of Jacobi forms of index 0, we obtainby convoluting with a Jacobi form of index m a family of Jacobi forms also of index m . Aclear description of the Wiles trick for classical modular forms can be found in Hida’s book[Hid93, § I = Λ,thanks to base change property: J ord N ( I ) ∼ = J ord N (Λ) ⊗ Λ I as I is Λ-free, cf. [Hid95, argument onlast line of p.149 and two first lines of p.150].Denote by E ( ψ ) the Λ-adic Eisenstein series defined in [Hid93, p. 198] , where ψ = ω a isan even Dirichlet character with 0 ≤ a < p −
1, and ω the Teichm¨uller character. -ADIC FAMILIES OF JACOBI FORMS 11 Consider a Jacobi form F ∈ J k ′ ,m (Γ ( p s ) , χ ), and the product F · E ( ψ ) inside Λ[[ q, ζ ]]. Inparticular, we get the product of F and the classical Eisenstein series specialized at weight k as a classical Jacobi form:(1) F · E ( ψ )( u k − ∈ J k + k ′ ,m (Γ ( p s ) , χψω − k ) . Let us write F · E ( ψ )( X ) = P r ≤ nm c n,r ( X ) q n ζ r . Define the convolution product of F and E ( ψ ) giving a family over Λ (cf. [Hid95, p.200]): F ∗ E ( ψ )( X ) := X r ≤ nm c n,r (cid:16) u − k ′ χ − ( u ) X + ( u − k ′ χ ( u ) − − (cid:17) q n ζ r , that is, we have the specialization F ∗ E ( ψ )( χ − ( u ) u k − k ′ − ∈ J k,m (Γ ( p s ) , χψω − k ) for all k > k ′ as well as for k = 0 from Equation (1) just above, and hence the desired Λ-adic familyof Jacobi forms in the sense of Definition 2.9. Remark . Thanks to the work of Guerzhoy [Gue95], [Gue94], the above computation canbe generalized using the more involved Jacobi-Eisenstein series of non-trivial index but thisgives rise to a non-trivial shift in the index of the resulting family.3. Λ -adic theta liftings
The results of this section are similar to that obtained by Stevens and Hida in the contextof the theta lifting to spaces of half-integral weight modular forms [Ste94], [Hid95]. Similartechniques have already been employed in the context of Jacobi forms by Guerzoy in [Gue95],[Gue94], [Gue00a] and [Gue00b].3.1.
Theta liftings.
In this section, we recall the connection between classical Jacobi formsand modular forms via theta liftings, following [EZ85] and [MR00].We first set up some general notation, which will be used throughout the paper. For anyinteger M ≥
1, any Dirichlet character χ : ( Z /M Z ) × → C × and any k ∈ Z , denote by S k (Γ ( M ) , χ ) the complex vector space of elliptic cuspforms of weight k , level Γ ( M ) andcharacter χ . If M = S · T and χ is a character modulo T , we denote by J cusp k,S ( T, χ ) thecomplex vector space of Jacobi forms of weight k , level Γ ( T ), character χ and index M (see[EZ85, Ch. I, §
1] for the definition). If T = 1 (so the character χ is trivial), we denote thecorresponding spaces simply by S k (Γ ( M )) and J cusp k,M , respectively. For f ∈ S k (Γ ( M ) , χ )and φ ∈ J cusp k,S ( T, χ ) we have the respective Fourier expansion f ( z ) = P n ≥ a n q n and Fourier-Jacobi expansion φ ( τ, z ) = P r,n c ( n, r ) q n ζ r , where q = e πiτ and ζ = e πiz and the Fourier-Jacobi expansion is taken over all pairs ( n, r ) of integers such that r − Sn < m ) and T J ( m ) be the Hecke operators at m ∈ N acting onthe space of elliptic modular forms S k (Γ ( M ) , χ ) and Jacobi forms J cusp k,S ( T, χ ), respectively.If ℓ is a prime number, which is not coprime to the level (in the case of elliptic modularforms) respectively, not coprime to the index or the level (in the case of Jacobi forms), wedenote the operators by U( ℓ ) and U J ( ℓ ) respectively. We recall the description of the action ofHecke operators on Jacobi forms in some special cases (cf. [EZ85, Ch. I § § φ ( τ, z ) = P r,n c ( n, r ) q n ζ r be a form in J cusp k,S ( T, χ ). Write ( φ | T J ( m ))( τ, z ) = P r,n c ∗ ( n, r ) q n ζ r .If m is coprime with M , we have (cf. [EZ85, Thm. 4.5]) c ∗ ( n, r ) = X d | m (cid:18) Dd (cid:19) d k − c (cid:18) m nd , mrd (cid:19) . Moreover, if m = ℓ is a prime number dividing T , then For any ring R , let P k − ( R ) denote the R -module of homogeneous polynomials in 2 variablesof degree k − R , equipped with the right action of the semigroup M ( R )defined by the formula(2) ( F | γ )( X, Y ) = F ( aX + bY, cX + dY )for γ = (cid:0) a bc d (cid:1) ∈ M ( R ). Let V k − ( R ) denote the R -linear dual of P k − ( R ), equipped with theleft M ( R )-action induced from the action on P k − ( R ).For each integer ∆, let Q ∆ be the set of integral quadratic forms Q = [ a, b, c ] = ax + bxy + cy of discriminant ∆ and, for any integer M ≥ ρ , let Q M, ∆ ,ρ denote the subsetof Q ∆ consisting of integral binary quadratic forms Q = [ a, b, c ] of discriminant ∆ such that b ≡ ρ (mod 2 M ) and a ≡ M . Let Q M, ∆ ,ρ be the subset of Q M, ∆ ,ρ consisting of formswhich are Γ ( M )-primitive, i.e., those Q ∈ Q M, ∆ ,ρ which can be written as Q = [ M a, b, c ]with ( a, b, c ) = 1. Those sets are equipped with the natural right action of SL ( Z ) describedin Equation (2).Let Q χ D ( Q ) be the generalized genus character attached to D defined in [GKZ87,Ch. I, Sec. 1]. The character χ D : Q M, ∆ ,ρ → {± } depends on ( M, ∆ , ρ ), and sometimeswe will denote it by χ ( M, ∆ ,ρ ) D to stress this dependency, or simply χ D as above if ( M, ∆ , ρ )is understood. We recall the characterization of this character given in [GKZ87, Prop. 1].If Q = ℓ · Q ′ for some form Q ′ ∈ Q M, ∆ ,ρ , then we define it as χ D ( Q ) = (cid:0) D ℓ (cid:1) · χ D ( Q ′ ), soit is enough to define it on Γ ( M )-primitive forms. Fix Q ∈ Q M, ∆ ,ρ . If ( a/M, b, c, D ) = 1,then pick any factorization M = S · T with S > T > n coprime with D represented by the quadratic form [ a/S, b, cT ]: then χ D ( Q ) := (cid:0) D n (cid:1) . Otherwise i.e., if( a/M, b, c, D ) = 1, then we set χ D ( Q ) := 0.For Q ∈ Q M, ∆ ,ρ , denote by C Q the oriented geodesic path in the upper half plane attachedto Q joining the points r Q and s Q in P ( Q ), whose construction is described in [Ste94, § D = Div ( P ( Q )) be the group of degree-zero divisors on rational cusps of the complexupper half plane H , equipped with the left action of the semigroup M ( Q ) by fractional lineartransformations. If A is a Z [ M ( R )]-module the group of homomorphisms Hom( D , A ) isequipped with a standard left action of M ( R ) defined for γ ∈ M ( R ) and φ ∈ Hom( D , A )by ( γ · φ )( d ) = γ · φ ( d ). If Γ is a congruence subgroup of SL ( Z ) and A is a Z [Γ]-module asabove, let Symb Γ ( A ) = Hom Γ ( D , A )be the group of A -valued Γ-invariant modular symbols. As a general notation, if m is amodular symbol in Symb Γ ( A ), we write m { r → s } ∈ A for the value of m on the divisor s − r ∈ D .We now recall the construction of the theta lifting of the modular form f to the complexvector space of Jacobi forms, following [GKZ87, Ch. II].Fix an integer M ≥
1, a divisor T | M , and a Dirichlet character χ : ( Z /T Z ) × → C × . Put S = M/T , and assume that (
S, T ) = 1. Let f be a normalized cuspform of positive evenweight 2 k and level Γ ( M ), where M ≥ χ . To f we mayassociate the modular symbol ˜ I f ∈ Symb Γ ( M ) ( V k − ( C )) by the integration formula˜ I f { r → s } ( P ) = 2 πi Z sr f ( z ) P ( z, dz. Definition 3.1.
A index S pair is a pair ( D, r ) of integers consisting of a negative discriminant D of an integral quadratic form Q = [ a, b, c ] such that D ≡ r mod 4 S . An index S pair issaid to be fundamental if D is a fundamental discriminant. -ADIC FAMILIES OF JACOBI FORMS 13 Fix a fundamental index S pair ( D , r ). For any index S pair ( D, r ) let F ( D,r ) D ,r ( S, T ) be theset of integral binary quadratic forms Q = [ a, b, c ] modulo the right action of Γ ( M ) describedin Equation (2), such that: • δ Q = b − ac = T D D ; • b ≡ − T r r mod 2 S ; • a ≡ ST .If T = 1, we simply write F ( D,r ) D ,r ( M ) for F ( D,r ) D ,r ( M, Q is arepresentative of a class in F ( D,r ) D ,r ( S, T ), then Q belongs to Q S,T D D, − T r r . In particular, wemay consider the genus character Q χ D ( Q ) with χ D = χ ( S,T D D, − T r r ) D , for all classes Q in F ( D,r ) D ,r ( S, T ). We may also define a character Q = [ a, b, c ] χ ( Q ) = χ ( c ) on the set F ( D,r ) D ,r ( S, T ).For this paper, we only need to consider the theta liftings studied in [MR00] in two particularsituations: when χ is primitive and when χ is trivial and T is a prime number. We make thedefinition in the two cases separately. If χ is primitive, define˜ I f ( D, r ) = X Q ∈F ( D,r ) D ,r ( S,T ) χ ( Q ) · χ D ( Q ) · Z γ Q f ( z ) Q ( z, k − dz := X Q ∈F ( D,r ) D ,r ( S,T ) χ ( Q ) · χ D ( Q ) · ˜ I f { r Q → s Q } ( Q k − )while if χ is trivial, and T is a prime number, define˜ I f ( D, r ) := T k − X Q ∈F ( D,r ) D ,r ( S,T ) χ ( Q ) · χ D ( Q ) · ˜ I f { r Q → s Q } ( Q k − ) − (cid:18) D T (cid:19) · X Q ∈F ( T D,Tr ) D ,r ( ST , χ ( Q ) · χ D ( Q k − ) · ˜ I f { r Q → s Q } ( Q ) Remark . The definition in [MR00] when χ is trivial looks slightly different, since the secondsum is taken over quadratic forms modulo Γ ( N p ); however, each γ ∈ Γ ( N p ), γ Γ ( N p )gives the same contribution to the above sum and we get the formula displayed above bytaking into account the different powers of T used here and in [MR00].For D = r − Sn , put ˜ c f ( n, r ) = ˜ I f ( D, r ). Then ˜ S ( χ ) D ,r ( f ) = P r,n ˜ c f ( n, r ) q n ζ r belongsto J cusp k +1 ,S ( T, χ ). The map f ˜ S ( χ ) D ,r ( f ) from S k (Γ ( M ) , χ ) to J cusp k +1 ,S ( T, χ ) thus obtainedis called the ( D , r )-theta lifting to the space of Jacobi forms (cf. [EZ85, Ch. II], [MR00,Sec. 3]). The matrix (cid:0) − (cid:1) normalizes Γ ( M ) and hence induces an involution on thespace of modular symbols Symb Γ ( M ) ( V k − ( C )); for each ε ∈ {±} , we denote by ˜ I εf the ε -eigencomponents of ˜ I f with respect to this involution. It is known that there are complexperiods Ω εf such that the I εf = ˜ I εf Ω εf belong to Symb Γ ( M ) ( V k − ( F f )), where F f is the extensionof Q generated by the Fourier coefficients of f . These periods can be chosen so that thePetersson norm h f, f i equals the product Ω + f · Ω − f ; note that the Ω εf are well-defined only upto multiplication by non-zero factors in F × f .Let Q ∈ F ( D,r ) D ,r ( S, T ). It follows easily from the definition of the generalized genus characterthat χ D ( Q ) = χ D ( Q | ι ) (it is enough to note that if the integer n is represented by thequadratic form [ a/S, b, cT ] with the integers x and y , then it is represented by the quadratic form [ a/S, − b, cT ] with the integers x and − y ). Also, one can easily check that r Q = − s Q | ι and s Q = − r Q | ι , and therefore (cid:16) ( ˜ I f | ι ) { r Q → s Q } (cid:17) ( Q k − ) = ˜ I f {− r Q → − s Q } (cid:16) ( Q | ι ) k − (cid:17) = − ˜ I f { r Q | ι → s Q | ι } (cid:16) ( Q | ι ) k − (cid:17) . Finally, it is immediate that χ ( Q ) = χ ( Q | ι ). Combining these observations, we find that˜ I f ( D, r ) = X Q ∈F ( D,r ) D ,r ( S,T ) χ ( Q ) · χ D ( Q ) · ˜ I − f { r Q → s Q } ( Q k − ) . Normalize the theta lifting by S ( χ ) D ,r ( f ) = ˜ S ( χ ) D ,r ( f )Ω − f . We thus obtain a map S ( χ ) D ,r : S k (Γ ( M ) , χ ) −→ J cusp k +1 ,S ( T, χ )such that, if we write out the Fourier expansion S ( χ ) D ,r ( f ) = X r − Sn< c f ( n, r ) q n ζ r then for D = r − Sn we have(3) c f ( n, r ) = X Q ∈F ( D,r ) D ,r ( S,T ) χ ( Q ) · χ D ( Q ) · I − f { r Q → s Q } ( Q k − ) . In particular, if the Fourier coefficients of f belong to a certain ring O ⊆ ¯ Q , then the same istrue for the Fourier-Jacobi coefficients of S ( χ ) D ,r ( f ). If χ = is the trivial character, then wewill simply write S D ,r for S ( ) D ,r . The map S ( χ ) D ,r are equivariant with respect to the actionof Hecke operators i.e., S ( χ ) D ,r ( f | T( m )) = S ( χ ) D ,r ( f ) | T J ( m ).3.2. Universal ordinary modular symbols.
Recall that O is the valuation ring of a finitefield extension of Q p . Let Cont( Z p , O ) be the O -module of O -valued continuous functionson Z p , and let Step( Z p , O ) be the O -submodule of Cont( Z p , O ) consisting of locally constantfunctions. Let ˜ D = Hom Z p (Step( Z p ) , O )be the group of O -valued measures on Z p ; we can extend in a unique way any µ ∈ ˜ D to afunction µ : Cont( Z p , O ) → O . Adopting a standard convention, for ϕ a continuous functionon Z p , we denote R Z p ϕ ( x ) dµ the value µ ( ϕ ); if χ X is the characteristic function of X ⊆ Z p , wewrite R X ϕ ( x ) dµ ( x ) for R Z p ϕ ( x ) · χ X ( x ) dµ ( x ). Let D denote the O -submodule of ˜ D consistingof those O -valued measures which are supported on the set of primitive vectors X = ( Z p ) ′ in Z p . The O -modules ˜ D and D are equipped with the action induced by the action of the groupGL ( Z p ) on Z p by ( x, y ) ( ax + by, cx + dy ) for γ = (cid:0) a bc d (cid:1) ∈ GL ( Z p ). The O -module D isalso equipped with a structure of O [[ Z × p ]]-module induced by the scalar action of Z × p on X (cf.[Ste94, § § D is also equipped with a structure of Λ-module.Let Γ ( p Z p ) and Γ ( p Z p ) denote the subgroups of GL ( Z p ) consisting of matrices which arerespectively upper triangular and congruent to a matrix of the form (cid:0) ∗ (cid:1) modulo p .The group Symb Γ ( N ) ( D ) is equipped with an action of Hecke operators, and we denote by W = Symb ordΓ ( N ) ( D )the ordinary subspace of Symb Γ ( N ) ( D ) for the action of the U p -operator; see [GS93, (2.2),(2.3)] for details and more accurate definitions. For any Λ-algebra R , write W R = W ⊗ Λ R . -ADIC FAMILIES OF JACOBI FORMS 15 We fix as in § N ≥
1, a prime number p ∤ N , p ≥
5, a primitive branch R of the Big Hida Hecke algebra acting on modular forms of tame level Γ ( N ) and a primitiveHida familiy f ∞ ( κ ) = P n ≥ a n ( κ ) q n ∈ R [[ q ]]. If κ is an arithmetic point of X ( R ) of signature( χ, k ), and wild level p s , then we have a Γ s -equivariant homomorphism ρ κ : D → V k − ( C p )defined by the formula ρ κ ( µ )( P ) = Z Z p × Z × p χ ( y ) · P ( x, y ) dµ ( x, y ) . The homomorphism ρ κ gives rise to an homomorphism, denoted by the same symbol, ρ κ : W R −→ Symb Γ ( N ) ( V k − ( C p )) . For Φ ∈ W R , we put Φ κ = ρ κ (Φ).Fix an arithmetic point κ . By [Ste94, Thm. 5.5], there exists Φ κ ∈ Symb Γ ( N ) ( D R ) suchthat(4) Φ κ ( κ ) = λ ( κ ) · I − f κ with λ ( κ ) ∈ R κ such that λ ( κ ) = 1. When κ is understood, we will simply write Φ for Φ κ .3.3. Λ -adic liftings. For any Λ-algebra R , we write D R = D ⊗ Λ R . We switch back to thenotation used for Hida families over GL . Recall that e X ( R ) is the O -module of continuous O -linear homomorphisms from e R to ¯ Q p , where e R = R ⊗ Λ e Λ and the Λ-algebra e Λ is equal to Λas an O -module, but the structure of Λ-algebra is given by the map σ ( t ) = t on 1 + p Z p . Wehave the notion of arithmetic points e X arith ( R ) and that of signature ( χ, k ) of an arithmeticpoint ˜ κ in e X arith ( R ); the effect of π on an arithmetic point ˜ κ is to double the signature: if ˜ κ has signature ( χ, k ), then π (˜ κ ) has signature ( χ , k ).The following result is essentially [Ste94, Lemma (6.1)], with minor modifications whichare left to the reader. Lemma 3.3.
Fix a positive integer k , a positive integer s , and a quadratic form Q = [ a, b, c ] such that p s | a , p s | b , p ∤ c . There exists a unique morphism j Q,k : D R → e R of R -modules characterized by the following property: For any pair of points κ ∈ X arith ( R ) and ˜ κ ∈ X arith ( e R ) of signature ( χ , k ) and ( χ, k ) respectively, satisfying the conditions (1) π (˜ κ ) = κ ; (2) 2 k ≡ k (mod p − ; (3) the wild level of χ is p s ;we have ˜ κ ( j Q,k ( µ )) = χ ( Q ) · ρ κ ( µ ) (cid:0) Q k − (cid:1) . We now prove a couple of auxiliary lemmas on quadratic forms. We denote by P ( D,r ) D ,r ( N, p s )the subset of F ( D,r ) D ,r ( N, p s ) consisting of classes represented by quadratic forms [ a, b, c ] with p ∤ c . Note that if s ≥
1, and [ a, b, c ] ∈ P ( D,r ) D ,r ( N, p s ), then p s | b : this is because p s | a and p s | b − ac , which implies that p s | b . Lemma 3.4.
The map P ( D,r ) D ,r ( N, p s ) → P ( Dp s − ,rp s − ) D ,r ( N, p ) which takes [ a, b, c ] [ a, b, c ] is a bijection.Proof. The map is clearly well defined.We first show the surjectivity. For this, we need to show that we may choose a representative[ a, b, c ] for any class in P ( Dp s − ,rp s − ) D ,r ( N, p ) so that p s | a , and for this it is enough to showthat p s | b because p s | δ Q and p ∤ c . So fix Q = [ a, b, c ] a representative of a class in P ( Dp s − ,rp s − ) D ,r ( N, p ). Let a ∈ Z such that a ≡ − b/ N cp (mod p s ) (note that p | b because p s | b − ac and p | a , so, since p ∤ c , − b/ cp is a p -adic integer) and consider the matrix γ = (cid:0) Nap (cid:1) . Then Q | γ = [ a ′ b ′ c ′ ] with b ′ = b + 2 N cpa ≡ p s ). Since γ ∈ Γ ( N p ),this shows that we can change representatives of any element in P ( Dp s − ,rp s − ) D ,r ( N, p ) so that p s | a , and therefore this is an element in P ( D,r ) D ,r ( N, p s ), proving the surjectivity. We now showthat the above map is injective. Take two quadratic forms [ a, b, c ] and [ a ′ , b ′ , c ′ ] representingclasses in P ( D,r ) D, ,r ( N, p s ) and assume they are Γ ( N )-equivalent. Recall that p ∤ cc ′ . Sincea set of representatives of Γ ( N ) modulo Γ ( N p s ) is given by the matrices γ i = (cid:0) iN (cid:1) for i = 0 , . . . , p s −
1, we see that, up to changing the representative [ a ′ , b ′ , c ′ ], there exists0 ≤ i < p s such that [ a ′ , b ′ , c ′ ] = [ a, b, c ] | γ i , and therefore a ′ = a + biN + ci N . Since p s | a and p s | a ′ , we then see that p s | iN ( b + ciN ). Suppose now i = 0, and write t = ord p ( i ).Then p s − t | b + ciN because p ∤ N . So b ≡ − ciN (mod p s − t ), and therefore b ≡ c ( iN ) (mod p s ). On the other hand, p s | b , and therefore, since p ∤ N c , we get p s | i , which is notpossible. This proves that i = 0, and therefore [ a, b, c ] and [ a ′ , b ′ , c ′ ] are Γ ( N p s )-equivalent,proving the injectivity of the map. (cid:3) Recall the fixed arithmetic point κ ∈ X arith ( R ) of signature ( χ , k ), and the modularsymbol Φ κ attached to κ in Equation (4). For Q in F ( D,r ) D ,r ( N, p ) put D Q = s Q − r Q . Let˜ κ ∈ X arith ( e R ) be such that the pair ( κ , ˜ κ ) satisfies the condition of Lemma 3.3. Extend j Q,k to a map j Q,k : D R κ → e R ˜ κ . Fix a fundamental index N pair ( D , r ). For any index N pair ( D, r ), define(5) c n,r := X Q ∈P ( D,r ) D ,r ( N,p ) χ D ( Q ) · j Q,k (Φ κ ( D Q )) . Here χ D = χ ( N,DD p , − rr p ) D . Definition 3.1.
The ( D , r )-family of Shintani lifts centered at κ is the formal power seriesin S ( κ ) D ,r in e R κ [[ q, ζ ]] defined by(6) S ( κ ) D ,r = X D = r − Nn< c n,r q n ζ r . When κ is fixed, we will simply write S D ,r in place of S ( κ ) D ,r to lighten the notation. Remark . In the definition above the coefficients c n,r in Equation (5) depend on the choiceof D , r and κ , but we do not make explicit this dependence to make the notation simpler;similar remarks will apply to similarly defined coefficients. However, we keep here and in thefollowing the dependence on these quantities in the notation for the Λ-adic families themselves,such as S ( κ ) D ,r .Let κ be fixed from now on, and write Φ and S D ,r for Φ κ and S ( κ ) D ,r . For any ˜ κ ∈ X ( e R )write S D ,r (˜ κ ) = ˜ κ ( S D ,r ) for the specialization at ˜ κ . As the name suggests, S ( κ ) D ,r inDefinition 3.1 interpolates in families Shintani lifts of classical forms in Hida families, as thefollowing theorem shows. Theorem 3.6.
Let κ ∈ X arith ( R ) be an arithmetic point of signature ( χ , k ) and let ˜ κ ∈ e X arith ( R ) be an arithmetic point of signature ( χ, k ) be such that π (˜ κ ) = κ . Let χ = ω k − ψ bethe character of ˜ κ , so that χ is the character of κ , and assume that the conductor of χ is p s for some integer s ≥ . Then S D ,r (˜ κ ) = λ ( κ ) · S ( χ ) D ,r ( f κ ) | T J ( p ) − s . -ADIC FAMILIES OF JACOBI FORMS 17 Proof.
We first observe that, combining Lemma 3.3 and Equation (4), for all Q ∈ P ( D,r ) D ,r ( N, p s )we have˜ κ ( j Q,k (Φ κ ( D Q ))) = χ ( Q ) · Φ κ ( D Q )( k ) (cid:16) Q k − (cid:17) = λ ( k ) · χ ( Q ) · I − f κ { r Q → s Q } (cid:16) Q k − (cid:17) . Next, note that if the quadratic form Q = [ a, b, c ] has c = 0, then χ ( Q ) = 0 and thesummand corresponding to Q does not appear in Equation (3). Therefore, the sum in Equation(3) apparently over F ( D,r ) D ,r ( N, p s ) is really only over P ( D,r ) D ,r ( N, p s ).We now prove the theorem. If s = 1, the term T J ( p ) − s disappears so this is an immediateconsequence of the previous two observations, so let us assume that s > F = P n,r a n,r q n ζ r ∈ A [[ q, ζ ]], where A is any ring, and m ≥ F | T mJ ( p ))( q, ζ ) = X n,r a p m n,p m r q n ζ r . The formulas recalled in Section 3.1 show that if F is a Jacobi form in J cusp k +1 ,N ( p s , χ ), thenthe operator just defined coincides with the Hecke operator T mJ ( p ), justifying the abuse ofnotation. We have S D ,r (˜ κ ) | T s − J ( p ) = λ ( κ ) · X Q ∈P ( Dp s − ,rps − D ,r ( N,p ) χ ( Q ) · χ D ( Q ) · I − f κ { r Q → s Q } (cid:16) Q k − (cid:17) . Lemma 3.4 shows that there exists a common set of representatives for P ( Dp s − ,rp s − ) D ,r ( N, p )and P ( D,r ) D ,r ( N, p s ), and therefore the sum in the right hand side of the last displayed equalityis λ ( k ) · S ( χ ) D ,r ( f κ ). It might be also useful to notice that we may alternatively view χ D asa function on P ( Dp s − ,rp s − ) D ,r ( N, p ) or P ( D,r ) D ,r ( N, p s ), since in both cases χ D is defined as χ ( N,D Dp s , − r rp s ) D . We therefore get the equality of power series S D ,r (˜ κ ) | T s − J ( p ) = λ ( κ ) · S ( χ ) D ,r ( f κ ) . The Hecke operator T J ( p ) acts on S ( χ ) D ,r ( f κ ) as multiplication by the p -adic unit a p ( κ ), andthe theorem follows then applying T − sJ ( p ) to the above formula. (cid:3) We now study the specialization of the family of Jacobi forms S D ,r to arithmetic pointswith trivial character. Theorem 3.7.
Let κ ∈ X arith ( R ) be an arithmetic point of signature ( , k ) , let ( D , r ) bea fundamental index N pair such that p ∤ D . Let ˜ κ be an arithmetic point of ˜ X arith ( R ) suchthat π (˜ κ ) = κ . Then, for any index N pair ( D, r ) such that p ∤ D , satisfying the relation D = r − N n for an integer n , we have c n,r (˜ κ ) = λ ( κ ) · c f κ ( n, r ) , where c f κ ( n, r ) is the Fourier-Jacobi coefficients of S ( ) D ,r ( f κ ) .Proof. We first note that the result is clear if ˜ κ has signature ( χ, k ) with χ = , since then χ ( Q ) = 0 for all non-primitive forms. So assume that χ = . As in the proof of Theorem 3.6,combining Lemma 3.3 and Equation (4) we get c n,r (˜ κ ) = λ ( κ ) · X Q ∈P ( D,r ) D ,r ( N,p ) χ D ( Q ) · I − f κ { r Q → s Q } (cid:16) Q k − (cid:17) . Since p ∤ D D , the set F ( D,r ) D ,r ( N, p ) is the disjoint union of the two sets P ( D,r ) D ,r ( N, p ) and thesubset N ( D,r ) D ,r ( N, p ) of F ( D,r ) D ,r ( N, p ) consisting of non-primitive forms. Therefore, X Q ∈P ( D,r ) D ,r ( N,p ) χ D ( Q ) · I − f κ { r Q → s Q } (cid:16) Q k − (cid:17) == X Q ∈F ( D,r ) D ,r ( N,p ) χ D ( Q ) · I − f κ { r Q → s Q } (cid:16) Q k − (cid:17) − X Q ∈N ( D,r ) D ,r ( N,p ) χ D ( Q ) · I − f κ { r Q → s Q } (cid:16) Q k − (cid:17) . Since χ D ( p · Q ) = (cid:16) D p (cid:17) · χ D ( Q ) if Q is not primitive, and r p · Q = r Q and s p · Q = s Q , thesecond summand of the last displayed formula is equal to p k − (cid:18) D p (cid:19) · X Q ∈S ( D,r ) D ,r ( N,p ) χ D ( Q ) · I − f κ { r Q → s Q } (cid:16) Q k − (cid:17) . where S ( D,r ) D ,r ( N, p ) = { Q/p : Q ∈ N ( D,r ) D ,r ( N, p ) } . The set S ( D,r ) D ,r ( N, p ) consists then of forms[ a, b, c ] modulo Γ ( N p ) satisfying the three conditions b − ac = D D , b ≡ − r r (mod N )and a ≡ N p ), and therefore this is F ( D,r ) D ,r ( N p, (cid:3)
We now relate the coefficients c n,r (˜ κ ) with the theta lift of oldforms appearing in the Hidafamilies. More precisely, let κ be an arithmetic point of even weight 2 k > f κ is then p -old, and there exists a newform f ♯κ of level Γ ( N )and trivial character whose p -stabilization is f κ ; in other words, we have the relation f κ ( z ) = f ♯κ ( z ) − p k − a p ( κ ) f ♯κ ( pz ) . If k = 2 and the character is trivial, then either f κ is the p -stabilization of a form f ♯κ as above,or f κ is a newform of level Γ ( N p ). We need a couple of technical lemmas on quadratic forms.
Lemma 3.8.
Let ( D , r ) be a fixed fundamental index N pair. For any index N pair ( D, r ) ,there exists a system of representatives R ( Dp ,rp ) D ,r ( N ) for P ( Dp ,rp ) D ,r ( N ) consisting of forms [ a, b, c ] with p ∤ c and p | b .Proof. Fix a p -primitive form Q = [ a, b, c ] representing a class in P ( D,r ) D ,r ( N, p ). We first claimthat, up to a change of representatives, we can assume that p | b . To show this, note that if p ∤ b then p ∤ a because p | = b − ac . Choose any integer β such that β ≡ − b/ a (mod p ), andlet γ = (cid:0) β (cid:1) . Then Q | β = [ a ′ , n ′ , c ′ ] with p | c ′ . So we may assume that the representative Q = [ a, b, c ] is chosen so that p | b . If now p | c , then p | a , because otherwise [ a, b, c ] is not p -primitive. Since p | b and p | b − ac , we see that p | c . Take two integers α and β such that − N α + p β = 1 and consider the matrix γ = (cid:0) pβ αN p (cid:1) . Then Q | γ has the requiredproperty that p | b and p ∤ c . (cid:3) Lemma 3.9.
Let R ( Dp ,rp ) D ,r ( N ) be fixed as in Lemma 3.8 and ( D , r ) a fundamental index N pair and ( D, r ) an index N pair. The canonical map [ a, b, c ] [ a, b, c ] induces a bijection P ( D,r ) D ,r ( N, p ) → R ( Dp ,rp ) D ,r ( N ) .Proof. The map is clearly well defined. We show that it is a bijection. We first show thesurjectivity. It is clearly enough to show that each p -primitive form in the target comes froma (necessarily p -primitive) form in the source via the map in the statement. By Lemma 3.8it is enough to show that each quadratic form [ a, b, c ] with p ∤ c and p | b , representing a -ADIC FAMILIES OF JACOBI FORMS 19 class in F ( Dp ,rp ) D ,r ( N ), also represents a class in F ( D,r ) D ,r ( N, p ), namely, p | a . This is clear: thediscriminant b − ac of the quadratic form [ a, b, c ] is divisible by p , b is divisible by p and p ∤ c .We show the injectivity. Take two primitive quadratic forms [ a, b, c ] and [ a ′ , b ′ , c ′ ] represent-ing classes in F ( D,r ) D, ,r ( N, p ) and assume they are Γ ( N )-equivalent. Since these are p -primitiveforms, p ∤ cc ′ . A set of representatives of Γ ( N ) modulo Γ ( N p ) is given by the matrices γ i = (cid:0) iN (cid:1) for i = 0 , . . . , p −
1. So, up to changing the representative [ a ′ , b ′ , c ′ ], there exists0 ≤ i < p such that [ a ′ , b ′ , c ′ ] = [ a, b, c ] | γ i , and therefore a ′ = a + biN + ci N . Since p | a and p | a ′ , we then see that p | iN ( b + ciN ). Suppose now i = 0. Then p | b + ciN because p ∤ N . So b ≡ − ciN (mod p ). However, p | b but p ∤ N c , so we get p | i , which isnot possible because i < p . This proves that i = 0, and therefore [ a, b, c ] and [ a ′ , b ′ , c ′ ] areΓ ( N p )-equivalent, proving the injectivity of the map. (cid:3)
Theorem 3.10.
Assume that f κ is the p -stabilization of f ♯κ , and let ( D , r ) be a fundamentalindex S pair. Let ˜ κ be an arithmetic point of ˜ X arith ( R ) of signature ( , k ) , such that π (˜ κ ) = κ ,an arithmetic point in X arith ( R ) of signature ( , k ) . Then, for any index N pair ( D, r ) suchthat p ∤ D , satisfying the relation D = r − N n for an integer n , we have c n,r (˜ κ ) = λ ( κ ) · a p ( κ ) − p k − (cid:16)(cid:16) D p (cid:17) + (cid:16) Dp (cid:17)(cid:17) a p ( κ ) − p k − a p ( κ ) · c f ♯κ ( n, r ) , where c f ♯κ ( n, r ) is the Fourier-Jacobi coefficient of S ( ) D ,r ( f ♯κ ) .Proof. As in the proof of Theorem 3.6, combining Lemma 3.3 and Equation (4) we get c n,r (˜ κ ) = λ ( κ ) · X Q ∈P ( D,r ) D ,r ( N,p ) χ D ( Q ) · I − f κ { r Q → s Q } (cid:16) Q k − (cid:17) . By definition, Z s Q r Q f κ ( z ) Q ( z, k − dz = Z s Q r Q f ♯κ ( z ) Q ( z, k − dz − p k − a p ( κ ) Z s Q r Q f ♯κ ( pz ) Q ( z, k − dz. Changing variable z z/p , we have Z s Q r Q f ♯κ ( pz ) Q ( z, k − dz = 1 p · Z s Qp r Qp f ♯κ ( z ) Q p ( z, k − dz where Q p = [ a/p , b/p, c ]. We therefore obtain the equality(7) I − f κ { r Q → s Q } (cid:16) Q k − (cid:17) = I − f ♯κ { r Q → s Q } (cid:16) Q k − (cid:17) − p k − a p ( κ ) I − f ♯κ { r Q p → s Q p } (cid:16) Q k − p (cid:17) . To compute c n,r (˜ κ ) we need to take the sum over all forms Q ∈ P ( D,r ) D ,r ( N, p ) in Equation (7).We begin by computing the sum over all Q ∈ P ( D,r ) D ,r ( N, p ) of the first summand in the righthand side of Equation (7). Suppose first that p ∤ D . Then p ∤ D D and the set F ( Dp ,rp ) D ,r ( N )is the disjoint union of the two sets P ( Dp ,rp ) D ,r ( N ) and p · P ( D,r ) D ,r ( N ). Therefore, identifying P ( D,r ) D ,r ( N, p ) with R ( Dp ,rp ) D ,r ( N ) as above, we see that X Q ∈P ( D,r ) D ,r ( N,p ) χ D ( Q ) · I − f ♯κ { r Q → s Q } (cid:16) Q k − (cid:17) = = X Q ∈F ( Dp ,rp ) D ,r ( N ) χ D ( Q ) · I − f ♯κ { r Q → s Q } (cid:16) Q k − (cid:17) − X Q ∈ p ·P ( D,r ) D ,r ( N ) χ D ( Q ) · I − f ♯κ { r Q → s Q } (cid:16) Q k − (cid:17) . Since χ D ( p · Q ) = (cid:16) D p (cid:17) · χ D ( Q ) if Q is primitive, and r p · Q = r Q and s p · Q = s Q , the secondsummand of the last displayed formula is equal to p k − (cid:16) D p (cid:17) c f ♯κ ( n, r ), and we get X Q ∈P ( D,r ) D ,r ( N,p ) χ D ( Q ) · I − f ♯κ { r Q → s Q } (cid:16) Q k − (cid:17) = c f ♯κ ( np , rp ) − p k − (cid:18) D p (cid:19) c f ♯κ ( n, r ) . If p | D , then the above formula still holds, since in this case χ D ( Q ) = 0 when Q is not p -primitive, and therefore the sum in Equation (7) reduces to c f ♯κ ( np , rp ). Finally, using theformulas for the Hecke operator T J ( p ) in Sec. 3.1, we obtain the relation a p ( κ ) · c f ♯κ ( n, r ) = c f ♯κ ( np , rp ) + (cid:18) Dp (cid:19) p k − c f ♯κ ( n, r ) . We now compute the sum over all Q ∈ P ( D,r ) D ,r ( N, p ) of the second summand in Equation(7). View Q ∈ P ( D,r ) D ,r ( N, p ) as an element of R ( Dp ,rp ) D ,r ( N ) using Lemma 3.9. For each such Q ,the form Q p has discriminant D D , is p -primitive (because p ∤ D D ), and b ≡ r r (mod N ),so it belongs to P ( D,r ) D ,r ( N ). We claim that the subset R ( D,r ) D ,r ( N ) = { Q p | Q ∈ P ( D,r ) D ,r ( N, p ) } of P ( D,r ) D ,r ( N ) thus obtained forms a system of representatives for F ( D,r ) D ,r ( N ). To show this,first note that P ( D,r ) D ,r ( N ) = F ( D,r ) D ,r ( N ): since p ∤ D D , an integral quadratic form may havediscriminant D D only if it is p -primitive. Moreover, the map [ a, b, c ] [ p a, pb, c ] takes theset P ( D,r ) D ,r ( N ) to P ( Dp ,rp ) D ,r ( N ), and is clearly an inverse of the map [ a, b, c ] [ a/p , b/p, c ]defined above, thus proving the claim. Since χ D ( Q ) = χ D ( Q p ), we have X Q ∈P ( D,r ) D ,r ( N,p ) χ D ( Q p ) · I − f ♯κ { r Q p → s Q p } (cid:16) Q k − p (cid:17) = X Q ∈F ( D,r ) D ,r ( N ) χ D ( Q ) · I − f ♯κ { r Q → s Q } (cid:16) Q k − (cid:17) . The term on the right hand side is c f ♯κ ( n, r ).Putting everything together, we get the formula c n,r (˜ κ ) = λ ( k ) · (cid:18) a p ( κ ) · c f ♯κ ( n, r ) − p k − (cid:18)(cid:18) D p (cid:19) + (cid:18) Dp (cid:19)(cid:19) c f ♯κ ( n, r ) − p k − a p ( κ ) c f ♯κ ( n, r ) (cid:19) from which the result follows. (cid:3) Remark . The results described in this section can be used to link Stark-Heegner points([Dar01], [BD09a]; see also [LV14] and [LMH20]) to p -adic derivatives of Jacobi-Fourier coef-ficients of p -adic families of Jacobi forms, as done in [DT08] and [LM18] for p -adic families ofhalf-integral weight modular forms. Details are left to the interested reader.4. A pointer to [LN19]In [LN19], we consider the compatibility of the Big Heegner points construction of B.Howard of [How07] with the formation of Λ-adic Jacobi forms, in the spirit of the classicalGross-Kohnen-Zagier theorem cf. [GKZ87] which links Heegner points (or more generally,Heegner cycles in higher weight) with the theta lift of a weight 2 modular form in the spaceof Jacobi forms. More precisely, choose a index N pair ( D , r ) such that p is split in D ; -ADIC FAMILIES OF JACOBI FORMS 21 then the generating series naturally constructed from Big Heegner points and the index N pair ( D , n ) is the formal series H D ,r = X n,r α n,r q n ζ r in R [[ q, ζ ]], where α n,r are more precisely defined as follows. Let Sel( Q , T † ) be the GreenbergSelmer group of the self-dual twist T † of Hida’s Big Galois representation T . Under theassumptions of [LN19], this is an R -module of rank 1. Fix a generator Z of the torsion-freequotient of Sel( Q , T † ), and let K := Frac( R ). The K -vector space Sel K ( Q , T † ) := Sel( Q , T † ) ⊗ R K is one-dimensional, and therefore in Sel K ( Q , T † ) we may write Z How
D,r = α n,r · Z , where Z How
D,r ∈ Sel( Q , T † ) are Howard’s Big Heegner points, and α n,r are a priori elements in K ;moreover, by our choice of Z , we actually have that α n,r ∈ R . Loc. cit. links the coefficients α n,r explicitly with Fourier coefficients c f ♯κ ( n, r ) associated to f ♯κ as in Theorem 3.10, at leastin small connected neighborhoods of even weights (see [LN19, Rmk 5.3] for a discussion of ourperspective over the whole weight space). We briefly point out why the Fourier coefficients c n,r (˜ κ ) associated to f κ indeed vary p -adically, even though f ♯κ itself does not. Indeed, for2 k >
2, the term − p k − (cid:16)(cid:16) D p (cid:17) + (cid:16) Dp (cid:17)(cid:17) a p ( κ ) − p k − a p ( κ ) of Theorem 3.10 is always a p -adic unit. It then follows that the coefficients c n,r (˜ κ ) and c f ♯κ ( n, r ) are equal, up to non-zero fudge factors not depending on n and r . Upon dividingby the expression for c n ,r (˜ κ ) on both sides, these fudge factors therefore entirely disappear.The results on p -adic variation contained in [LN19] can therefore be expressed in terms of c n,r (˜ κ ) instead of c f ♯κ ( n, r ), although in a slightly more restrictive way. References [BD09a] M. Bertolini and H. Darmon,
The rationality of Stark-Heegner points over genus fields of real qua-dratic fieldseegner points over genus fields of real quadratic fields , Anna. Math. (2009), 343–369.[BD09b] Massimo Bertolini and Henri Darmon,
The rationality of Stark-Heegner points over genus fields ofreal quadratic fields , Ann. of Math. (2) (2009), no. 1, 343–370. MR 2521118[Boy15] Hatice Boylan,
Jacobi forms, finite quadratic modules and Weil representations over number fields ,Lecture Notes in Mathematics, vol. 2130, Springer, Cham, 2015, With a foreword by Nils-PeterSkoruppa. MR 3309829[Cana] Luca Candelori,
The Algebraic Functional Equation of Riemann’s Theta Function , to appear at Ann.Inst. Fourier.[Canb] ,
The Transformation Laws of Algebraic Theta Functions , preprint arXiv:1609.04486.[Can14] ,
Metaplectic stacks and vector-valuedmodular forms of half-integral weight , Ph.D. Thesis(2014).[Dar01] H. Darmon,
Integration on H p × H and arithmetic applications , Ann. Math. (2001), 589–639.[DR73] P. Deligne and M. Rapoport, Les sch´emas de modules de courbes elliptiques , 143–316. Lecture Notesin Math., Vol. 349. MR 0337993[DT08] H. Darmon and G. Tornar`’ia,
Stark–heegner points and the shimura correspondence , CompositioMathematica (2008), no. 5, 1155–1175.[EZ85] Martin Eichler and Don Zagier,
The theory of Jacobi forms , Progress in Mathematics, vol. 55,Birkh¨auser Boston, Inc., Boston, MA, 1985. MR 781735[GKZ87] B. Gross, W. Kohnen, and D. Zagier,
Heegner points and derivatives of L -series. II , Math. Ann. (1987), no. 1-4, 497–562. MR 909238[GS93] Ralph Greenberg and Glenn Stevens, p -adic L -functions and p -adic periods of modular forms , Invent.Math. (1993), no. 2, 407–447. MR 1198816[Gue94] P.I. Guerzhoy, An approach to the p -adic theory of Jacobi forms , Internat. Math. Res. Notices (1994),no. 1, 31–39. MR 1255251 [Gue95] , Jacobi-Eisenstein series and p -adic interpolation of symmetric squares of cusp forms , Ann.Inst. Fourier (Grenoble) (1995), no. 3, 605–624. MR 1340946[Gue00a] P. Guerzhoy, On p -adic families of Siegel cusp forms in the MaaßSpezialschar , J. Reine Angew. Math. (2000), 103–112. MR 1762957[Gue00b] P.I. Guerzhoy, Jacobi forms and a p -adic L -function in two variables , Fundam. Prikl. Mat. (2000),no. 4, 1007–1021. MR 1813009[Hid86a] Haruzo Hida, Galois representations into GL ( Z p [[ X ]]) attached to ordinary cusp forms , Invent.Math. (1986), no. 3, 545–613. MR 848685[Hid86b] , Iwasawa modules attached to congruences of cusp forms , Ann. Sci. ´Ecole Norm. Sup. (4) (1986), no. 2, 231–273. MR 868300[Hid88] , On p -adic Hecke algebras for GL over totally real fields , Ann. of Math. (2) (1988),no. 2, 295–384. MR 960949[Hid93] , Elementary theory of L -functions and Eisenstein series , London Mathematical Society Stu-dent Texts, vol. 26, Cambridge University Press, Cambridge, 1993. MR 1216135[Hid95] , On Λ -adic forms of half integral weight for SL(2) / Q , Number theory (Paris, 1992–1993),London Math. Soc. Lecture Note Ser., vol. 215, Cambridge Univ. Press, Cambridge, 1995, pp. 139–166. MR 1345178[How07] Benjamin Howard, Variation of Heegner points in Hida families , Invent. Math. (2007), no. 1,91–128. MR 2264805[Ibu12] Tomoyoshi Ibukiyama,
Saito-Kurokawa liftings of level N and practical construction of Jacobi forms ,Kyoto J. Math. (2012), no. 1, 141–178. MR 2892771[Kra91] J. Kramer, A geometrical approach to the theory of Jacobi forms , Compositio Math. (1991), no. 1,1–19.[Kra95] , An arithmetic theory of Jacobi forms in higher dimensions , J. Reine Angew. Math. (1995), 157–182. MR 1310957[LM18] Matteo Longo and Zhengyu Mao,
Kohnen’s formula and a conjecture of Darmon and Tornar´ıa ,Trans. Amer. Math. Soc. (2018).[LMH20] Matteo Longo, Kimball Martin, and Yan Hu,
Rationality of darmon points over genus fields of non-maximal orders , Annales math´ematiques du Qu´ebec (2020), no. 1, 173–195.[LN13] Matteo Longo and Marc-Hubert Nicole, The Λ -adic Shimura-Shintani-Waldspurger correspondence ,Doc. Math. (2013), 1–21. MR 3035767[LN19] , On the p -adic variation of the Gross-Kohnen-Zagier Theorem , Forum Math. (2019), no. 4,1069–1084.[LV11] Matteo Longo and Stefano Vigni, Quaternion algebras, Heegner points and the arithmetic of Hidafamilies , Manuscripta Math. (2011), no. 3-4, 273–328. MR 2813438[LV14] ,
The rationality of quaternionic Darmon points over genus fields of real quadratic fields , Int.Math. Res. Not. IMRN (2014), no. 13, 3632–3691. MR 3229764[MB90] Laurent Moret-Bailly,
Sur l’´equation fonctionnelle de la fonction thˆeta de Riemann , CompositioMath. (1990), no. 2, 203–217. MR 1065206[MR00] M. Manickam and B. Ramakrishnan, On Shimura, Shintani and Eichler-Zagier correspondences ,Trans. Amer. Math. Soc. (2000), no. 6, 2601–2617. MR 1637086[MRV93] M. Manickam, B. Ramakrishnan, and T. C. Vasudevan,
On Saito-Kurokawa descent for congruencesubgroups , Manuscripta Math. (1993), no. 1-2, 161–182. MR 1247596[Nek06] Jan Nekov´aˇr, Selmer complexes , Ast´erisque (2006), no. 310, viii+559. MR 2333680[Ram06] Nick Ramsey,
Geometric and p -adic modular forms of half-integral weight , Ann. Inst. Fourier (Greno-ble) (2006), no. 3, 599–624. MR 2244225[Ste94] Glenn Stevens, Λ -adic modular forms of half-integral weight and a Λ -adic Shintani lifting-adic Shintani lifting