The Bloch-Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms
TThe Bloch–Okounkov theorem for congruencesubgroups and Taylor coefficients of quasi-Jacobi forms
Jan-Willem M. van Ittersum ∗ February 26, 2021
Abstract
There are many families of functions on partitions, such as the shifted symmetricfunctions, for which the corresponding q -brackets are quasimodular forms. We extendthese families so that the corresponding q -brackets are quasimodular for a congruencesubgroup. Moreover, we find subspaces of these families for which the q -bracket is amodular form. These results follow from the properties of Taylor coefficients of strictlymeromorphic quasi-Jacobi forms around rational lattice points.
1. Introduction
Denote by P the set of all partitions of integers. For a large class of functions f : P → C the q -bracket , which is defined as the formal power series (cid:104) f (cid:105) q := (cid:80) λ ∈ P f ( λ ) q | λ | (cid:80) λ ∈ P q | λ | ∈ C [[ q ]] , is a quasimodular form for SL ( Z ) (here | λ | is the integer λ is a partition of). It is, therefore,natural to raise the following two questions:(I) Given a congruence subgroup Γ ≤ SL ( Z ), is there an (even larger) class of functionsfor which the q -bracket is a quasimodular form for Γ?(II) What is the class of functions for which the q -bracket is a modular form (and not justa quasimodular form)?Some (partial) answers to these two questions are being recalled below, but in this paperwe explain how to answer these questions by studying a different question of independentinterest: ∗ Email : [email protected],Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, The Netherlands,Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. a r X i v : . [ m a t h . N T ] F e b III) What is the modular or quasimodular behaviour of the Taylor coefficients of meromor-phic quasi-Jacobi forms?To illustrate how these questions are related, consider the shifted symmetric functions Q k ,for k ≥ Q k ( λ ) = β k + 1( k − ∞ (cid:88) i =1 (cid:0) ( λ i − i + ) k − − ( − i + ) k − (cid:1) , (1)where (cid:80) ∞ k =0 β k z k − = e z/ e z − . By the celebrated Bloch–Okounkov theorem, for every homoge-neous polynomial in these functions, the q -bracket is a quasimodular form for SL ( Z ). Moreprecisely, this result follows from the fact that (cid:104) W ( z ) · · · W ( z n ) (cid:105) q is certain meromorphicquasi-Jacobi form F n (defined by Definition 1.7), where W λ ( z ) = ∞ (cid:88) i =1 e (( λ i − i + ) z ) (Im( z ) < , e ( x ) = e π i x ) (2)is the generating series of the functions Q k [BO00, Zag16]. The study of the Taylor coefficientsof F n around rational values of z , . . . , z n yields an answer to Question (I), whereas a detaileddescription of the (quasi)modular transformation of the Taylor coefficients of F n around z i = 0answers Question (II). The Fourier coefficients of the functions F n where studied in [BM15],and the Taylor coefficients of certain functions closely related to holomorphic quasi-Jacobiforms were studied in [Bri18]. However, the Taylor coefficients of F n or of a more generalmeromorphic quasi-Jacobi form have not been studied before.We now give an short overview of the results in the literature as well as in the presentpaper. It should be noted that the answer to the third question can be read independentlyfrom the applications to functions on partitions. Given a ∈ Q , k ≥ λ ∈ P , let Q k ( λ, a ) = β k ( a ) + 1( k − ∞ (cid:88) i =1 (cid:0) e ( a ) λ i − i ( λ i − i + ) k − − e ( a ) − i ( − i + ) k − (cid:1) , (3)where (cid:80) k ∈ Z β k ( a ) (2 π i z ) k − = e ( z/ e ( z + a ) − and e ( x ) = e π i x . The main properties satisfied bythese functions are a consequence of the fact that (cid:88) k ≥ Q k ( λ, a ) z k − = e ( − a ) W λ ( z + a ) . and that F n ( z , . . . , z n ) = (cid:104) W ( z ) · · · W ( z n ) (cid:105) q is a quasi-Jacobi form. Up to a constant, thesefunctions Q k ( · , a ) have been considered before in [EO06] for a = and in [Eng17] for all a ∈ Q . It was shown that a suitably adapted q -bracket of any polynomial in these functions,excluding the function Q ( · , a ), is quasimodular for Γ ( N ) for some N .In this work we will not change the q -bracket, nor exclude any of the functions (3), andnevertheless prove the following result for the algebra Λ ∗ ( N ), contained in Q P , given byΛ ∗ ( N ) := Q (cid:2) Q k ( · , a ) | k ≥ , a ∈ { , N , . . . , N − N } (cid:3) (4)and graded by assigning weight k to Q k ( · , a ). Given N ≥
1, write (2 , N ) for gcd(2 , N ).2 heorem 1.1.
Let k ∈ Z , N ≥ and M = (2 , N ) N . For f ∈ Λ ∗ ( N ) of weight k , the q -bracket (cid:104) f (cid:105) q is a quasimodular form of weight k for w M Γ( M ) w − M , where w M denotes theFricke operator.Remark . The occurrence of the group w M Γ( M ) w − M in the theorem can be interpreted inat least two ways. First of all, equivalently f as in the theorem has the property that (cid:104) f (cid:105) q M is a quasimodular form of level M , where q M := q /M . Secondly, as Γ ( M ) ≤ w M Γ( M ) w − M ,it follows that (cid:104) f (cid:105) q is a quasimodular form for Γ ( M ).Moreover, the definition of M indicates that the behaviour of Q k ( a ) is different when thenumerator of a is divisible by 2. We will see that this can also by explained by the n -pointfunctions F n : they are quasi-Jacobi forms of half-integral index. (cid:52) The following theorem gives a refinement of Theorem 1.1 giving us q -brackets (or quotientsof q -brackets) that are quasimodular forms on Γ ( N ) rather than only on the much smallergroup w M Γ( M ) w − M . Theorem 1.3.
Let N ≥ . Given k i ∈ Z , a i ∈ N Z for i = 1 , . . . , n , denote a = a + . . . + a n and Q k ( · , a ) = Q k ( · , a ) · · · Q k n ( · , a n ) . Then, • If a ∈ Z , then (cid:104) Q k ( · , a ) (cid:105) q is a quasimodular form for Γ ( N ) ; • If a (cid:54)∈ Z , then (cid:104) Q k ( · , a ) (cid:105) q (cid:104) Q ( · , a ) (cid:105) q is a quasimodular form for Γ ( N ) .Remark . Let a ∈ Q . The function (cid:104) Q ( · , a ) (cid:105) − q , equal to Θ( a ), is a so-called Klein form ;see, e.g., [KL81]. In particular, (cid:104) Q ( · , a ) (cid:105) q is ill-defined for a ∈ Z . (cid:52) Theorem 1.1 should be compared with the results in [GJT16], where the authors considercertain functions Q ( p ) k (see (27), where we define Q ( m ) k also for composite m ) in the contextof studying p -adic analogues of the shifted symmetric functions (1). They show that the q -bracket of these functions is quasimodular for Γ ( p ) and suggest that it is likely that productsof these functions also have quasimodular q -brackets for the same group. We will see thatthe functions Q ( m ) k , as well as their products, are in fact elements of Λ ∗ ( m ). Therefore, for allodd m Theorem 1.1 implies that q -brackets of products of these functions are quasimodularfor Γ ( m ). That these q -brackets are indeed quasimodular for the bigger group Γ ( m )is the content of the next theorem. Write Λ ( N ) for the Q -graded algebra generated by thefunctions Q ( m ) k for all m | N . Theorem 1.5.
Let N ≥ and k ∈ Z . For all homogeneous f ∈ Λ ( N ) of weight k thefunction (cid:104) f (cid:105) q is a quasimodular form of weight k for Γ ( N ) . It should be noted that the above theorems hold true in a greater generality. For example,the so-called hook-length moments introduced in [CMZ18] and studied in the context of har-monic Maass forms for a congruence subgroup in [BOW20], also have natural generalisationsobtained by studying their corresponding n -point functions. Similarly, the moment functionsand their generalisations in [Itt20b] can equally well be generalised to congruence subgroups.Therefore, we will state and prove the above results in Section 3.1 in a more general settingthat allows application to the hook-length moments and moment functions.3 .2. When is the q -bracket modular? In order to illustrate the main ideas, consider again the Bloch–Okounkov algebra Λ ∗ = Λ ∗ (1).For all k ≥
0, let h k ∈ Λ ∗ be given by h k ( λ ) = (cid:98) k (cid:99) (cid:88) r =0 Q ( λ ) r Q k − r ( λ )2 r ( k − r − ) r r ! (( x ) n = x ( x + 1) · · · ( x + n − . We show that the function h k satisfies the following three properties:(i) The difference h k − Q k is divisible by Q ;(ii) The q -bracket (cid:104) h k (cid:105) q is a modular form (and not just a quasimodular form);(iii) For f ∈ Λ ∗ with (cid:104) f (cid:105) q modular and f − Q k divisible by Q , we have (cid:104) f (cid:105) q = (cid:104) h k (cid:105) q . Hence, one can think of the difference h k − Q k as a correction term for Q k with respectto the property of being a modular form under the q -bracket. By the third property thiscorrection term is unique up to elements in the kernel of the q -bracket. More generally, onehas the following result. Theorem 1.6.
For any algebra F of functions on partitions satisfying the conditions inSection 4.1, there exists a computable subspace M = M ( F ) ⊆ F such that (i) F = M ⊕ Q F ; (ii) (cid:104)M(cid:105) q ⊆ M ; (iii) (cid:104) Q F (cid:105) q ∩ M = { } , where M denotes the algebra of modular forms for SL ( Z ) . The algebra F = Λ ∗ is an example to which the above result applies. In particular,by using (i) inductively, each f ∈ Λ ∗ can uniquely be written as a polynomial in Q withcoefficients h i ∈ M , i.e., f = (cid:88) i ≥ h i Q i . As (cid:104) g (cid:105) q = 0 if and only if (cid:104) Q g (cid:105) q = 0 for g ∈ Λ ∗ , the following are equivalent:(a) (cid:104) f (cid:105) q is modular;(b) (cid:104) f (cid:105) q = (cid:104) h (cid:105) q ;(c) (cid:104) h i (cid:105) q = 0 for all i > h λ ∈ Λ ∗ defined in [Itt20a] form a basis for the space M (Λ ∗ ). The method ofproof in this work (i.e., using the quasi-Jacobi forms F n ) now allows us to generalise the aboveresult to different settings, whereas the results in [Itt20a] could not easily be generalised toother algebras than Λ ∗ . In Section 4, we answer the question when the q -bracket is modularnot only for the shifted symmetric functions, but also for their generalisation to congruencesubgroups and for the aforementioned moment functions.4 .3. Quasi-Jacobi forms and their Taylor coefficients Let τ ∈ h , the complex upper half plane, z ∈ C and write F = Z + , e ( x ) = e π i x , q = e ( τ ).Consider the Jacobi theta function and its derivatives θ ( τ, z ) = (cid:88) ν ∈ F ( − (cid:98) ν (cid:99) e ( νz ) q ν / θ ( r ) ( τ, z ) = (cid:16) π i ∂∂z (cid:17) r θ ( τ, z ) ( r ≥ , the Kronecker–Eisenstein series E k , for k ≥ E k ( τ, z ) = (cid:88) m,n ∈ Z z + mτ + n ) k , (5)and the Bloch–Okounkov n -point functions F n . Definition 1.7.
For all n ≥
1, let S n be the symmetric group on n letters and definethe Bloch–Okounkov n -point functions by F n ( τ, z , . . . , z n ) = (cid:80) σ ∈ S n V n ( τ, z σ (1) , . . . , z σ ( n ) ) , where the functions V n are defined recursively by V ( τ ) = 1 and n (cid:88) m =0 ( − n − m ( n − m )! θ ( n − m ) ( τ, z + . . . + z m ) V m ( τ, z , . . . , z m ) = 0 . (6)Do these functions have a common property? To answer this question, observe that thefirst two functions can be interpreted as generating functions of the Eisenstein series e k ,for k ≥ e k ( τ ) = (cid:88) mτ + n (cid:54) =0 (cid:48) mτ + n ) k , where the prime denotes the standard Eistenstein summation order in case k = 2. Namely,Θ( τ, z ) := θ ( τ, z ) θ (cid:48) ( τ,
0) = 2 π i z exp (cid:16) − (cid:88) m ≥ e m ( τ )2 m z m (cid:17) ,E k ( τ, z ) = 1 z k + ( − k (cid:88) m ≥ k/ (cid:18) m − k − (cid:19) e m ( τ ) z m − k . In particular, the Taylor (or Laurent) coefficients around z = 0 of these functions are poly-nomials in Eisenstein series, hence quasimodular forms—a property which is shared by thestronger notion of a Jacobi form. Just as e transforms as a quasi modular form, so do θ ( r ) and F n transform as quasi -Jacobi forms. Our answer to the question is that all these functionsare strictly meromorphic quasi-Jacobi forms, introduced in Section 2.4.Quasi-Jacobi forms transform comparable to quasimodular forms, as we explain now. Thereis a slash action (see Definition 2.2) on all functions ϕ : h × C n → C for all γ ∈ SL ( Z ) andfor all X ∈ M ,n ( Q ) (so actually for the action of their semidirect product SL ( Z ) (cid:110) M n, ( Q )).In case ϕ is a quasi-Jacobi form of weight k and index M , there exist quasi-Jacobi forms ϕ i, j ,indexed by a finite subset of Z ≥ × Z n ≥ , such that for all γ = (cid:0) a bc d (cid:1) and X = (cid:0) λµ (cid:1) ∈ M ,n ( Z )one has ( ϕ | k,M γ )( τ, z , . . . , z n ) = (cid:88) i, j ϕ i, j ( τ, z , . . . , z n ) (cid:16) ccτ + d (cid:17) i + | j | z j · · · z j n n , (7)5 ϕ | M X )( τ, z , . . . , z n ) = (cid:88) j ϕ , j ( τ, z , . . . , z n ) ( − λ ) j · · · ( − λ n ) j n , (8)together with similar formulas for each ϕ i, j | k,M γ and ϕ i, j | M X . Jacobi forms are quasi-Jacobiforms for which the right-hand side in both equations above equals ϕ .The quasi-Jacobi forms ϕ we study are strictly meromorphic , i.e., meromorphic such thatif z ∈ R n τ + R n is a pole of ϕ ( τ, · ) for some τ ∈ h , it is a pole for almost all τ ∈ h (seeSection 2.4). The Weierstrass ℘ -function is an example of a strictly meromorphic Jacobiform, but its inverse is not. In case n = 1, this condition is equivalent to the statement thatall poles of ϕ are torsion points z ∈ Q τ + Q . This is crucial in order to obtain mock modularforms as Fourier coefficients of meromorphic Jacobi forms (see [Zwe02, DMZ14]). For n > ϕ imply a more complicated restriction on thepositions of the poles. By studying the orbits of the action of SL ( Z ) on ( R ) n we prove thefollowing result. Theorem 1.8.
Let ϕ be a strictly meromorphic quasi-Jacobi form and τ ∈ h . Then allpoles z of ϕ ( τ, · ) lie in finite union of rational hyperplanes s z + . . . + s n z n ∈ pτ + q + L τ with s , . . . , s n ∈ Z and p, q ∈ Q / Z . Here L τ denotes the lattice Z τ + Z . In this work we determine conditions on the Taylor coefficients (or rather Laurent coeffi-cients in case we are expanding around a pole) of a meromorphic function ϕ : h × C n → C for it to be a (quasi-)Jacobi form. This is the technical result we need in order to answerQuestion I and II. In one direction, by the work of Eichler and Zagier [EZ85], it is knownthat for a Jacobi form the Taylor coefficients around rational lattice points are quasimodular,or equivalently that certain linear combinations ξ X(cid:96) of derivatives of these Taylor coefficientsare modular. For example, Θ is a weak Jacobi form, hence it satisfies(Θ | X | γ )( τ, z ) ∝ (Θ | γX )( τ, z )for all X = (cid:0) λµ (cid:1) ∈ M , ( Q ) and γ ∈ SL ( Z ), where the implicit multiplicative constant isa root of unity depending on X and γ . Hence, it follows that the Taylor coefficients of Θaround z = λτ + µ are quasimodular for some subgroup Γ X of SL ( Z ) consisting of γ forwhich γX − X ∈ M , ( Z ) (see (19)). In contrast, the weak quasi-Jacobi form Θ (cid:48) ( τ, z ) = θ (cid:48) ( τ,z ) θ (cid:48) ( τ, transforms as(Θ (cid:48) | X | γ )( τ, z ) ∝ (Θ (cid:48) | Xγ )( τ, z ) + czcτ + d (Θ | Xγ )( τ, z ) + λ (Θ | Xγ )( τ, z ) − λcτ + d (Θ | Xγ )( τ, z ) . Therefore, the Taylor coefficients of Θ (cid:48) + λ Θ around z = λτ + µ , rather than of Θ (cid:48) , arequasimodular forms for Γ X . The main result on Taylor coefficients of quasi-Jacobi forms isgiven by Theorem 2.42 and summarized in the following result. For simplicity, we assumethat s in the result above is always a standard basis vector, e.g., we allow ℘ ( z ) ℘ ( z + ),but we do not allow ℘ ( z − z ). 6 heorem 1.9. Let ϕ be a strictly meromorphic quasi-Jacobi form of weight k and index Q whose poles z lie on a finite collection of hyperplanes of the form z s ∈ pτ + q + L τ with s ∈ { , . . . , n } and p, q ∈ Q / Z . Then (i) for all X ∈ M n, ( Q ) and (cid:96) ∈ Z n the ‘Taylor coefficients’ g X (cid:96) ( ϕ ) , defined by Defini-tion 2.34, are quasimodular forms of weight k + | (cid:96) | for the group Γ X and satisfy thefunctional equations (22) and (23) . (ii) for all X ∈ M n, ( Q ) and m ∈ Z n the combinations ξ X m ( ϕ ) of the derivatives of g X (cid:96) ( ϕ ) ,defined by (21) , are modular forms of weight k + | m | for Γ X and satisfy the functionalequation (22) . Definition 2.34 and Equation (21) use the functions ϕ i, j in (7) and (8), which are uniquelydetermined by the quasi-Jacobi form ϕ . The results of Section 2.7 also show that the fourcollections of functions { ϕ } , { ϕ i, j } , { g X (cid:96) } and { ξ X m } determine each other in a computableway, and give explicit conditions on the collections { ϕ i, j } , { g X (cid:96) } and { ξ X m } that imply thatthey arise from a quasi-Jacobi form.In Section 2, we introduce meromorphic quasi-Jacobi forms and prove Theorem 1.9 andTheorem 1.8. The quasimodular transformation of the Taylor coefficients g X (cid:96) ( F n ) will thenimply the results on the Bloch–Okounkov theorem for congruence subgroups in Section 3.Moreover, in Section 4, we see that pulling back the functions ξ X m ( F n ) under the q -bracketleads to the construction of the functions h m for which the q -bracket is modular. Acknowledgement
I am very grateful for the encouragement and feedback received from both my supervisorsGunther Cornelissen and Don Zagier, especially taking into consideration the present diffi-culties to meet each other in person. Also, I want to thank Georg Oberdieck for introducingme to quasi-Jacobi forms.
2. Quasi-Jacobi forms
The definition of Jacobi forms in [EZ85] has been generalised in many ways. We providea generalization that incorporates several elliptic variables, characters, weakly holomorphicand meromorphic functions, and quasi-Jacobi forms [Lib11, Boy15, OP19]. In particular,the normalized Jacobi theta function Θ, the Kronecker–Eisenstein series E k and the Bloch–Okounkov n -point functions F n from the introduction will be examples throughout this paper.For τ ∈ h , the complex upper half plane, we write L τ = Z τ + Z . As is customary, weoften omit the dependence on the modular variable τ in any type of Jacobi form, e.g. wewrite Θ( z ) for Θ( τ, z ). We write a = ( a , . . . , a n ) for a vector of elements and we write | a | for a + . . . + a n . For vectors a and b , we write a b to denote (cid:81) i a b i i . Also, given X ∈ M ,n ( R ),we often write X = (cid:0) λµ (cid:1) with λ , µ ∈ R n and for γ ∈ SL ( Z ), we write γ = (cid:0) a bc d (cid:1) , γX = (cid:0) λ γ µ γ (cid:1) . In accordance with the action of SL ( Z ) on h , all actions of SL ( Z ) on other spaces are left actions. 7 .1. Strictly meromorphic Jacobi forms The definition of a strictly meromorphic Jacobi form ϕ is subtle, excluding many meromor-phic functions transforming as Jacobi forms. For example, the j -invariant, the reciprocal ofthe e Eisenstein series or ℘/ ∆, where ℘ is the Weierstrass ℘ -function and ∆ the modulardiscriminant, are all non-examples of strictly meromorphic Jacobi forms. Namely, first of all,although a strictly meromorphic Jacobi form is meromorphic we want its Taylor coefficientsin the elliptic variables to be holomorphic (rather than weakly holomorphic or meromor-phic) quasimodular forms. But with this not everything has been said, the definition is evenstricter: we require the poles of ϕ to be “constant” in the modular variable τ . Consider, forexample, the Weierstrass ℘ -function, which is an example of a strictly meromorphic Jacobiform. For every fixed z ∈ C (e.g., z = i) the function ℘ ( τ, z ) is a meromorphic function of τ ,as ϕ ( τ, z ) has a pole whenever τ is such that z ∈ L τ (e.g. τ = z = i). However, for λ, µ ∈ R ,the function ϕ ( τ, λτ + µ ) is holomorphic, unless both λ, µ ∈ Z . That is, all poles of ℘ ( τ, z )are given by z ∈ L τ . To the contrary, we will see that ℘ − is not a strictly meromorphicJacobi form, as the poles of ℘ − are not “constant” in τ .Before introducing strictly meromorphic Jacobi forms, we first recall the Jacobi group andits action on (meromorphic) functions. Definition 2.1.
For all n ∈ N , the (discrete) Jacobi group Γ Jn of rank n is defined as the semi-direct product Γ Jn = SL ( Z ) (cid:110) M ,n ( Z ) with respect to the left action of SL ( Z ) on M ,n ( Z ).That is, an element of Γ Jn is a pair ( γ, X ) with γ ∈ SL ( Z ) , X ∈ M ,n ( Z ) and satisfies thegroup law ( γ, X )( γ (cid:48) , X (cid:48) ) = ( γγ (cid:48) , γ (cid:48) X + X (cid:48) ).Let M ∈ M n ( Q ). We often make use of the associated quadratic form Q M and a bilinearform B M , given by Q M ( z ) = z M z t , B M ( z , z (cid:48) ) = z M z (cid:48) t . Definition 2.2.
Given a meromorphic function ϕ : h × C n → C , k ∈ Z and M ∈ M n ( Q ), forall ( γ, X ) ∈ Γ Jn we let(i) ( ϕ | k,M γ )( τ, z ) := ( cτ + d ) − k e (cid:16) − c Q M ( z ) cτ + d (cid:17) ϕ (cid:16) aτ + bcτ + d , z cτ + d (cid:17) (cid:0) γ = (cid:0) a bc d (cid:1)(cid:1) ;(ii) ( ϕ | M X )( τ, z ) := e ( Q M ( λ + µ )) e ( B M ( λ , λ τ + 2 z )) ϕ ( τ, z + λ τ + µ ) (cid:0) X = (cid:0) λµ (cid:1)(cid:1) . Moreover, we let ϕ | k,M ( γ, X ) := ( ϕ | k,M γ ) | M X , often omitting k and M from the notation. Remark . Given k ∈ Z and M ∈ M n ( Q ), the slash operator defines an action of Γ Jn of weight k and index M on the space of all meromorphic functions ϕ : h × C n → C . (cid:52) Given M ∈ M n ( Q ), for X, X (cid:48) ∈ M ,n ( Q ), we let ρ ( X ) e ( Q M ( λ ) − B M ( λ , µ ) + Q M ( µ )) , ζ X,X (cid:48) = e ( B M ( λ (cid:48) , µ ) − B M ( λ , µ (cid:48) )) . (9)By extending the slash action to the real Jacobi group in Appendix A, generalizing [EZ85,Theorem 1.4] to several variables and half-integral index, we obtain the following functionalequations. 8 roposition 2.4. Given a mermorphic function ϕ : h × C n → C , k ∈ Z and M ∈ M n ( Q ) ,for all X, X (cid:48) ∈ M ,n ( R ) and γ ∈ SL ( Z ) one has ρ ( − X ) ϕ | X | γ = ρ ( − γX ) ϕ | γ | γX and ρ ( − X ) ρ ( − X (cid:48) ) ϕ | X | X (cid:48) = ρ ( − X (cid:48) ) ρ ( − X ) ϕ | X (cid:48) | X = ρ ( − X − X (cid:48) ) ζ X,X (cid:48) ( ϕ | X + X (cid:48) ) . Classical modular forms are defined as the invariants for a certain group action of thespace Hol ( h ) of holomorphic functions in h satisfying a certain growth condition (that haveat most polynomial growth near the boundary). Definition 2.5.
Let Hol ( h ) be the ring of holomorphic functions ϕ of moderate growthon H , i.e. for all C > γ ∈ SL ( Z ) and x ∈ R one has ϕ ( γ ( x + i y )) = O ( e Cy ) as y → ∞ (where γ acts on H by M¨obius transformations).With the remarks at the beginning of this section in mind, we now define strictly mero-morphic Jacobi forms. A final subtlety in the definition below is coming from the fact thata meromorphic function in two or more variables always has points of indeterminacy (thinkof x/y near the origin, whose limiting value depends on the angle of approach). Points ofindeterminacy are not “generic”, and we exclude these points when we say for instance, thata certain function ϕ ( τ, z ) has its poles precisely on certain hyperplanes for all generic τ ∈ h . Definition 2.6.
Given n ≥
0, denote by Mer n the space of meromorphic functions ϕ : h × C n → C such that for all λ , µ ∈ R n either z = λ τ + µ is a pole of ϕ ( τ, · ) for all generic τ ∈ h or the function τ (cid:55)→ ϕ ( τ, λ τ + µ ) belongs to Hol ( h ). Moreover, given M ∈ M n ( Q ), denoteby Mer Mn the subspace of ϕ ∈ Mer n for which ϕ | M X ∈ Mer for all X ∈ M ,n ( Q ). Let Hol n and Hol Mn be the subspace in Mer n and Mer Mn , respectively, of holomorphic functions. Definition 2.7.
Let k ∈ Z and M ∈ M n ( Q ). A holomorphic , weak , or a strictly meromorphicJacobi form of weight k , index M and rank n for the Jacobi group Γ Jn is a function ϕ in Hol Mn ,Hol n or Mer Mn , respectively, that is invariant under the action Γ Jn of weight k and index M (i.e., ϕ | k,M g = ϕ for all g ∈ Γ Jn ). Remark . The space of Jacobi forms is trivial whenever 2 Q M is a non-integral quadraticform, or, equivalently, when m ij (cid:54)∈ Z or m ii (cid:54)∈ Z for i (cid:54) = j where M = ( m ij ). Namely,let ϕ be a Jacobi form and τ ∈ h fixed. If ϕ is non-zero of rank 1 it follows from the elliptictransformation law (Definition 2.2 (ii)) that the number of zeros minus the number of polesof z (cid:55)→ ϕ ( τ, z ) in any fundamental domain for the action of L τ on C is exactly 2 m , where M = ( m ) is the index of ϕ . For Jacobi forms of higher rank the integrability of 2 Q M followsby noting that for fixed µ , . . . , µ n ∈ C , functions of the form z (cid:55)→ ϕ ( τ, z, µ , µ , . . . , µ n ) and z (cid:55)→ ϕ ( τ, z, z, µ , . . . , µ n ) still satisfy the elliptic transformation law. (cid:52) Letting M be the 0-dimensional matrix, a holomorphic Jacobi form (of rank 0) is just amodular form. More interestingly, the Kronecker–Eisenstein series (5) for k ≥ E k ( τ, z ) = ( − k ( k − D k − y (cid:18) (cid:88) m ∈ Z yq m (1 − yq m ) (cid:19) (cid:16) y = e ( z ) , D y = y ∂∂y (cid:17) . ℘ -function and its derivative, that is ℘ = E − e and ℘ (cid:48) = − E (here E is defined by (5) using the same summation order as for e ). By [EZ85,Theorem 9.4] it follows that the space of weak Jacobi forms is given by J weak = C [ A, B, C, e , e ] / ( C − AB + 60 e A B + 140 e A ) , where A, B and C are equal to Θ , ℘ Θ and ℘ (cid:48) Θ respectively. Note that the relation C =4 AB − e A B − e A is equivalent to the relation for the Weierstrass ℘ -function. Thefollowing result, yielding an algebraic proof and extending [Lib11, Proposition 2.8] , gives allstrictly meromorphic Jacobi forms with only poles at the lattices points. The correspondingalgebra is free, as the relation between ℘ and ℘ (cid:48) can be used to express e in terms of thegenerators. As usual, we write m (instead of the matrix M = ( m )) for the index of a Jacobiform of rank 1. Proposition 2.9.
Let ϕ be a strictly meromorphic Jacobi form of rank and index m ∈ Z ≥ for which all poles z are lattice points z ∈ L τ . Then, ϕ ∈ C [ ℘, ℘ (cid:48) , e ] Θ m . Proof.
First, we show that f = ϕ Θ − m is a strictly meromorphic Jacobi form of index 0 withall poles z satisfying z ∈ L τ . This follows from the claim that Θ − is a strictly meromorphicJacobi form of index − /
2, weight 1 and with all the poles at the lattice points. In order toprove this claim, note that by the Jacobi triple productΘ = ( y / − y − / ) (cid:89) n ≥ (1 − yq n )(1 − y − q n )(1 − q n ) . It follows that Θ is a weak Jacobi form with all zeros at the lattice points z ∈ L τ . Moreover,for all X ∈ M ,n ( Q ) the function Θ | X does not vanish at infinity, from which the claimfollows.From now on, assume that ϕ is of index 0, i.e. that the function ϕ is an elliptic function.Write ϕ = ϕ + ϕ with ϕ and ϕ the even and odd part of ϕ respectively. For u, v ∈ R τ + R ,write u ∼ v if u ≡ v or u ≡ − v mod L τ . Then, for i = 0 , ϕ i = ( ℘ (cid:48) ) i (cid:89) j ( ℘ − ℘ ( u j ( τ ))) m j where m j ∈ Z and u j ( τ ) are representatives with respect to the above equivalence relationfor the zeros and poles of ϕ i outside L τ . As both ϕ and ϕ do not admit poles outside thelattice, it follows that m j >
0. Hence, ϕ is a polynomial in ℘ and ℘ (cid:48) where the coefficients arepolynomials in the functions ℘ ( u j ( τ )). By the modular transformation every such coefficientis a modular form for SL ( Z ), hence an element of C [ ℘, ℘ (cid:48) , e ]. Remark . Although the above result and many examples of strictly meromorphic Jacobiforms in the literature have only poles at z ∈ L τ , one easily constructs a strictly meromorphic The author states the result for
Jacobi forms , which obviously is not meant to be holomorphic Jacobi forms.Although not stated explicitly, we assume he refers to strictly meromorphic Jacobi forms with only polesat lattice points. ϕ is a Jacobi form with all poles at z ∈ L τ , then ϕ ( τ, z + ) + ϕ ( τ, z + τ ) + ϕ ( τ, z + τ + )is a Jacobi form for the same group, but now with the poles at , + τ and τ modulo thelattice L τ . (cid:52) In contrast to the space of (weakly) holomorphic Jacobi forms, the space of strictly mero-morphic Jacobi forms of given index and weight is not finite dimensional. However, the latterspace is not far from being finite dimensional. First of all, in contrast to the space of allstrictly meromorphic functions, the space of strictly meromorphic Jacobi forms is not a field.Moreover, the poles lie in a finite number of hyperplanes and after fixing finitely many suchhyperplanes to contain the poles, the vector space of strictly meromorphic Jacobi forms ofgiven index and weight is finite dimensional, as we will explain in this section.Given a meromorphic Jacobi form ϕ , we write P ϕ = { ( τ, z ) ∈ h × C n | ϕ is not holomorphic at ( τ, z ) } for the set of poles as well as points of indeterminacy of ϕ . We identify two points of P ϕ ifthey have same image under the projection h × C n (cid:16) M ,n ( R ) given by ( τ, λ τ + µ ) (cid:55)→ (cid:0) λµ (cid:1) .That is, we define an equivalence relation on P ϕ by saying that ( τ, z ) ∼ ( τ (cid:48) , z (cid:48) ) whenever,after writing z = λ τ + µ and z (cid:48) = λ (cid:48) τ + µ (cid:48) with λ , λ (cid:48) , µ , µ (cid:48) ∈ R n , one has λ = λ (cid:48) and µ = µ (cid:48) .We identify the quotient set Q ϕ with a subset of M ,n ( R ) by identifying a point of P ϕ withits image under the projection. From the definition of a strictly meromorphic Jacobi formwe obtain the factorisation h × Q ϕ (cid:39) P ϕ of P ϕ , given by ( τ, (cid:0) λµ (cid:1) ) (cid:55)→ ( τ, λ τ + µ ). Note that Q ϕ is invariant under translation by M ,n ( Z ).As an example of how the definition works, we first prove a simple consequence: Proposition 2.11.
Let ϕ be a strictly meromorphic Jacobi form of weight k and index .If ϕ is also a strictly meromorphic Jacobi form, then ϕ is constant.Proof. Let X ∈ M ,n ( R ) be given with X (cid:54)∈ Q ϕ and X (cid:54)∈ Q /ϕ . Write z ( τ ) = z = X ( τ, t .Then both ϕ ( τ, z ) and ϕ ( τ, z ) are holomorphic as a function of τ ∈ h . Hence, as a functionof τ ∈ h , both ϕ ( τ, z ) and ϕ ( τ, z ) do not admit any zeros. Similarly, it follows that both ϕ ( τ, z )and ϕ ( τ, z ) are holomorphic (as a function of τ ∈ h ) at the cusps, hence don’t admit any zeroat the cusps. Hence, ϕ ( τ, z ) doesn’t admit any zeros and poles on a compact set, so ϕ ( τ, z )is constant as a function of τ . As this holds for almost all X ∈ M ,n ( R ), we conclude that ϕ is globally constant.The following result, which is crucial for the sequel, tells us that the image of Q ϕ in thetorus M ,n ( R / Z ) consists of finitely many hyperplanes given by linear equations with rationalcoefficients. In other words, the following result strengthens Theorem 1.8 (the fact that thesecond conclusion is also true for quasi -Jacobi forms, follows immediately after introducingsuch functions (see Corollary 2.24)). 11 heorem 2.12. Let ϕ be a strictly meromorphic Jacobi form of weight k and index M , andlet P ϕ (cid:39) h × Q ϕ be the set of non-holomorphic points as above. Then, we have: (i) If X ∈ Q ϕ , then γX ∈ Q ϕ for all γ ∈ SL ( Z ) . (ii) There exist finitely many hyperplanes of the form s · X ∈ ( u, v ) with s ∈ Z n primitive (i.e., with coprime entries) and u, v ∈ Q / Z , such that X ∈ Q ϕ precisely if X lies on such a hyperplane.Proof. (i): Let X ∈ Q ϕ and γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ) be given. Write x ( τ ) = ( τ, X . Then, bythe modular transformation behaviour for γ − , it follows that ϕ (cid:16) dτ − c − bτ + a , z − cτ + a (cid:17) has a pole at z = x ( τ ) for generic τ ∈ h . Now, let τ (cid:48) = γ − τ . Writing z ( τ ) = λ τ + µ , wefind z ( γτ (cid:48) ) − cγτ (cid:48) + a = λ γτ (cid:48) + µ − cγτ (cid:48) + a = λ ( aτ (cid:48) + b ) + µ ( cτ (cid:48) + d ) = γ z ( τ (cid:48) ) . Hence, the function ϕ ( τ, z ) has a pole at z = γ x ( τ ) for generic τ ∈ h .(ii): Let X = (cid:0) λµ (cid:1) ∈ Q ϕ .First, we treat the case that the rank n is 1, in which case the set Q ϕ is discrete. Asproven in the next subsection, both λ and µ are rational. Hence, we can take s = 1 and( u, v ) = ( λ, µ ) mod Z .Next, let n ≥
2. The set Q ϕ being closed and ( n − Y ∈ M ,n ( R ) with Y (cid:54)∈ Q ϕ and which is bounded away from γX . Hence, by an approximationproperty proven in the next subsection, there are non-trivial s ∈ Z n , t ∈ Z and υ, ν ∈ Z with s · ( υ λ + ν µ ) = t . Note that the value of s , t, υ , and ν is a function of X , i.e., for all elements X ∈ Q ϕ there exists an element ( s , t, ( υ, ν )) ∈ Rel giving the relation, whereRel = ( Z n \{ } ) × Z × (cid:0) Z \{ (0 , } (cid:1) . For almost all X ∈ Q ϕ , the conditions of the implicit function theorem are satisfied. In thiscase (identifying M ,n ( R ) with C n via X (cid:55)→ x = ( τ, X ) there exist an open set U ⊂ C n − containing , an open neighbourhood V ⊂ C n of x , and a holomorphic function g : U → V ,such that z ∈ V is a pole precisely if z lies in the image of g . Let g i denote the componentfunctions of g , which take values in C (cid:39) R τ + R . Denote this isomorphism by ( π τ , ρ τ ). Forexample, taking τ = i, we find π i ( g i ) = Im( g i ) and ρ i ( g i ) = Re( g i ). For every element of U we find a relation of the form n (cid:88) i =1 s i ( u π τ ( g i ) + v ρ τ ( g i )) = t, (10)where ( s , t, ( u, v )) is an element of Rel. Recall that the set of zeros of a non-constant real-analytic function has measure 0. As π τ ( g i ) and ρ τ ( g i ) are real-analytic functions, either (10)holds for all elements of U , or it holds for a real subspace of U of measure 0. Now, note that12here are only countable elements in Rel, whereas countable many subspaces of measure 0do not cover U . Hence, we find a relation (10) which holds for all elements of U . By theCauchy–Riemann equations we can ‘upgrade’ this relation to the statement that s · z takes aconstant value uτ + v in R τ + R for all z ∈ g ( U ) (possibly after multiplying s by an integer).Without loss of generality we assume that gcd( s , . . . , s n ) = 1, if not we scale s , p and q appropriately.Now, by the Weierstrass preparation theorem, locally around x the set of poles is givenby k branches coming together, where each branch is an ( n − k equals the multiplicity of the pole at x (see[CJ12, Lemma 6.1] for the same argument in a different setting). By the previous we knowthat almost all (hence all) the elements in such a branch satisfy s · x = pτ + q . Write T for ( R τ + R ) / ( Z τ + Z ). By analytically extending such a local branch, we find that allsolutions z ∈ T of s · z = pτ + q are poles of ϕ ( τ, · ) as long as they are in the same connectedcomponent as x . Because gcd( s , . . . , s n ) = 1 the solution space of s · z = p i τ + q i in T isconnected, so that all solutions z of s · z ≡ pτ + q mod L τ are poles of ϕ ( τ, · ) for generic τ ∈ h .Moreover, ( u, v ) ∈ Q , as otherwise the solution space of s · z = uτ + v in T is not ( n − ϕ ( z ) by ϕ ( z ) Θ( s · z − uτ − v ), whichhas exactly the same poles as ϕ except for the poles which are zeros of s · z ≡ pτ + q mod L τ .By compactness of T it follows that one can restrict to only finitely many linear functions. Remark . Note that if s · X ∈ ( u, v ) is one of the equations determining Q ϕ , then s · X = ( u (cid:48) , v (cid:48) ) is another equation whenever ( u, v ) γ = ( u (cid:48) , v (cid:48) ) for some γ ∈ SL ( Z ). (cid:52) Corollary 2.14.
The vector space of strictly meromorphic Jacobi forms of some weight k ,index Q and poles ( τ, z ) only in the hyperplanes of the form s · z ∈ pτ + q + L τ with s ∈ Z n and p, q ∈ Q / Z is finite dimensional.Proof. This follows directly from the finite dimensionality of the space of weak Jacobi formsof fixed weight and index, together with the fact that the multiplicity at a pole is boundedby the weight k . That is, writing s i z ∈ p i τ − q i + L τ for the hyperplanes in the statement,indexed by i ∈ I , the function ϕ ( z ) (cid:81) i ∈ I Θ( s i · z − p i τ − q i ) k is a weak Jacobi form with indexand weight uniquely determined by ϕ and the s i . In this subsection, we prove the approximation properties that were used in the proof ofTheorem 2.12ii. That is, we prove a result indicating when given z , z (cid:48) ∈ R n τ + R n , thereexist γ ∈ SL ( Z ) such that γ z lies arbitrarily close to z (cid:48) . For X ∈ M m,n ( R ), write (cid:107) X (cid:107) forthe distance to the closest integer matrix, i.e. (cid:107) X (cid:107) := max i,j min (cid:96) ∈ Z | X i,j − (cid:96) | . roblem . Given
X, Y ∈ M ,n ( R ), find a condition on X and Y such that ∀ (cid:15) > ∃ γ ∈ SL ( Z ) : (cid:107) γX − Y (cid:107) < (cid:15) (11)holds. Suppose such a condition is satisfied, i.e. (11) is satisfied, we say Y is SL ( Z )- approximable by X .For example, in case n = 1 a necessary (but not sufficient) condition is given by X (cid:54)∈ M , ( Q ): Proposition 2.16.
Given
X, Y ∈ M , ( R ) and X (cid:54)∈ M , ( Q ) , then ∀ (cid:15) > ∃ γ ∈ SL ( Z ) : (cid:107) γX − Y (cid:107) < (cid:15). Proof.
Write X = (cid:0) λµ (cid:1) . First of all, if λ/µ is irrational, then the stronger statement that theorbit of (cid:0) λµ (cid:1) under SL ( Z ) lies dense in R holds (see e.g. [LN12]). Next, if both λ and µ areirrational, but with rational ratio, then (cid:18) a bc d (cid:19) (cid:18) λµ (cid:19) = µ (cid:32) a λµ + bc λµ + d (cid:33) . Note that the matrix on the right-hand side parametrizes N − ( Z ∧ Z ), where N is the denom-inator of λ/µ and Z ∧ Z denotes the subset of Z consisting of coprime integers. Therefore,the orbit of (cid:0) λµ (cid:1) under SL ( Z ) lies dense in R / Z . Finally, also if µ = 0, the orbit of (cid:0) λµ (cid:1) lies dense in R / Z whenever λ is irrational. Hence, if not both λ and µ are rational,then SL ( Z )-approximability of Y by X holds.If we replace SL ( Z ) by M m,n ( Z ), Problem 2.15 has an affirmative answer [Kro1884]. Theorem 2.17 (Kronecker’s theorem on Diophantine approximation) . For
X, Y ∈ M m,n ( R ) it holds that ∀ (cid:15) > ∃ γ ∈ M m ( Z ) : (cid:107) γX − Y (cid:107) < (cid:15) if and only if s ∈ Z n with s · X t ∈ Z m implies s · Y t ∈ Z m . (12)After identifying z with X ∈ M ,n ( R ), this is almost the result we are looking for, exceptthat we want to replace M ( Z ) by SL ( Z ). This question is being touched upon in for example[Dan75, Gui10, LN12], although the focus in the second and third work is a quantitativeversion in the size of γ whenever n = 1. The second work already hints on the fact that thecondition (12) should be altered if one replaces M ( Z ) by SL ( Z ). Namely, if λ, µ are coprimeintegers, then γ (cid:0) λµ (cid:1) is a vector of coprime integers for all γ ∈ SL ( Z ). Hence, if X = ( , ) t ,then (0 ,
0) is not in the orbit of X for SL ( Z ), although it is in the orbit of X for M ( Z ).Observe that in this case the smallest s ∈ Z ≥ for which s · (0 , ∈ Z does not equal thesmallest s ∈ Z ≥ for which s · X ∈ Z (i.e. 1 (cid:54) = 6), which is formalized in (14).For a generic point we show one has a Diophantine approximation theorem for SL ( Z ) forall n . 14 efinition 2.18. Given X ∈ M ,n ( R ), we say X is generic when for all υ, ν ∈ Z with υν (cid:54) = 0and for all s ∈ Z n with s (cid:54) = one has s · ( υ λ + ν µ ) (cid:54)∈ Z . In particular, X is generic when 1 , λ , . . . , λ n , µ , . . . , µ n are linearly independent over Q . Lemma 2.19 (Partial result on Diophantine approximation for SL ( Z )) . Let
X, Y ∈ M ,n ( R ) .Then, if X is generic one has ∀ (cid:15) > ∃ γ ∈ SL ( Z ) : (cid:107) γX − Y (cid:107) < (cid:15). (13) Conversely, if (13) holds, then for all s ∈ Z n for which s · X t ∈ Z one has that s · Y t ∈ Z and max (cid:110) N ∈ Z : s N ∈ Z n , s · X t N ∈ Z (cid:111) = max (cid:110) N ∈ Z : s N ∈ Z n , s · Y t N ∈ Z (cid:111) . (14) Proof.
Starting with the second part of the statement, note that if s ∈ Z n such that s · X t ∈ Z and s · Y t (cid:54)∈ Z , then for any γ ∈ SL ( Z ) one has (cid:107) γX − Y (cid:107) ≥ n (cid:107) s · ( γX ) t − s · Y t (cid:107) = 1 n (cid:107) s · Y t (cid:107) > . Moreover, suppose N ∈ Z and s ∈ Z n with N − s ∈ Z n , N − s · Y t ∈ Z and N − s · X t (cid:54)∈ Z .Write s (cid:48) = N − s . If γ ∈ SL ( Z ) is such that s (cid:48) · ( γX ) t ∈ Z , then s (cid:48) · X t ∈ Z , contradictingour assumption. Therefore, s (cid:48) · ( γX ) t ∈ ( N − Z ) \ Z . Hence, for any γ ∈ SL ( Z ) one has (cid:107) γX − Y (cid:107) ≥ n (cid:107) s · ( γX ) t − s · Y t (cid:107) = 1 n (cid:107) s · γX (cid:107) ≥ N n .
Next, let X be generic and (cid:15) >
0. In three steps we construct a γ ∈ SL ( Z ) for which (cid:107) γX − Y (cid:107) < (cid:15) . First we construct a matrix γ ∈ M ( Z ) for which (cid:107) γX − Y (cid:107) is small. Next,we find another γ ∈ M ( Z ) with the additional property that the entries in the first columnare coprime. Finally, in the third step, we find γ ∈ M ( Z ) of determinant equal to 1, i.e. γ ∈ SL ( Z ).First step. As X is generic, there is no s ∈ Z n such that s · X t ∈ Z n . Hence, by Kronecker’stheorem we find γ = (cid:16) a b c d (cid:17) ∈ M ( Z ) such that (cid:107) γX − Y (cid:107) < (cid:15) .Second step. Let k ∈ Z be such that (cid:107) ( ka − c + 1) µ (cid:107) < (cid:15), whose existence is guaranteed for generic X by Kronecker’s theorem. Then, γ := (cid:16) a b c d (cid:17) := γ + (cid:0) ka − c +1 0 (cid:1) satisfies (cid:107) γ X − Y (cid:107) ≤ (cid:107) γ X − Y (cid:107) + (cid:107) ( ka − c + 1) µ (cid:107) < (cid:15). As a = a and c = ka + 1, clearly ( a , c ) = 1.15hird step. Let ˜ b and ˜ d be given by the inverse Euclidean algorithm such that a ˜ d − c ˜ b = 1.We let (cid:96) ∈ Z be such that (cid:107) ( (cid:96)a + ˜ b − b ) λ + ( (cid:96)c + ˜ d − d ) µ (cid:107) < (cid:15), whose existence is again a consequence of X being generic. Then, let γ = γ + (cid:16) (cid:96)a +˜ b − b (cid:96)c + ˜ d − d (cid:17) . Invoking the triangle inequality again, we conclude (cid:107) γ X − Y (cid:107) ≤ (cid:15). Moreover, det γ = a ( (cid:96)c + ˜ d ) − c ( (cid:96)a + ˜ b ) = 1, which concludes the proof. Remark . Condition (14) is necessary for SL ( Z )-approximability of Y by X . Whetherthis condition also suffices, or, if not, how it should be strengthened to a necessary andsufficient condition remains an open problem. (cid:52) We weaken the notion of a strictly meromorphic Jacobi form even further by introducing thereal-analytic functions ν ( τ ) = 12i Im( τ ) , ξ ( τ, z ) = Im( z )Im( τ ) , which almost transform as a Jacobi form of index 0 and weight 1 and 2 respectively:( ν | , γ )( τ ) = ν ( τ ) − ccτ + d ( ξ | , γ )( τ, z ) = ξ ( τ, z ) − czcτ + d ( ν | X )( τ ) = ν ( τ ) ( ξ | X )( z ) = ξ ( z ) + λ. Definition 2.21.
Let k ∈ Z , M ∈ M n ( Q ). Denote by C a subspace of all strictly meromor-phic functions h × C n → C . An almost Jacobi form
Φ of analytic type C , weight k , index Q and rank n satisfies:(i) Φ ∈ C [ ν ( τ ) , ξ ( τ, z ) , . . . , ξ ( τ, z n )];(ii) Φ | g = Φ for all g ∈ Γ Jn .A quasi-Jacobi form ϕ is the constant term with respect to ν and ξ of an almost Jacobiform. If C equals Hol Mn , Hol n or Mer Mn , ϕ is a holomorphic , weak or strictly meromorphicquasi-Jacobi form , respectively.As a first example, quasimodular forms are quasi-Jacobi forms of rank n = 0.More interestingly, the functions E ( τ, z ) − π i ν ( τ ) , E ( τ, z ) + 2 π i ξ ( τ, z ) , E and E defined by regularizing the sum (5) as in [Wei99], are almost strictly mero-morphic Jacobi forms of index 0 and weight 2 and 1 respectively, so that E and E aremeromorphic quasi-Jacobi forms. Observe that E ( τ, z ) = − D z E ( τ, z ) E ( τ, z ) = (2 π i) D z log Θ( τ, z ) (cid:18) D z = 12 π i ∂∂z (cid:19) , which reminds one of e = 8 π D τ log η, (cid:18) η ( τ ) = q / (cid:89) n (1 − q n ) , D τ = 12 π i ∂∂τ (cid:19) . The strictly meromorphic quasi-Jacobi forms e and E play a central role as building blocksof quasi-Jacobi forms out of Jacobi forms. For convenience, we introduce the following alter-native normalizations e := 14 π e = 112 − (cid:88) m,r ≥ m q mr , A = 12 π i E . (15)Quasi-Jacobi forms are by construction strictly meromorphic, but they are not invariantunder the action of the Jacobi group. However, the fact that an almost Jacobi form is apolynomial in ν and ξ implies that a quasi-Jacobi form transforms “up to a polynomial cor-rection” as a Jacobi form, or, equivalently, that it is a polynomial in the strictly meromorphicquasi-Jacobi forms e ( τ ) and A ( τ, z ) with Jacobi forms as coefficients.Recall that for vectors a ∈ C n , b ∈ Z n we write a b = (cid:81) r a b r r . Proposition 2.22.
Two equivalent definitions for a strictly meromorphic quasi-Jacobi formare as follows: a strictly meromorphic quasi-Jacobi form is a function ϕ ∈ Mer Mn for whichthere exist a finite number of (1) functions ϕ i, j ∈ Mer Mn satisfying ( ϕ | γ )( τ, z ) = (cid:88) i, j ϕ i, j ( τ, z ) (cid:16) ccτ + d (cid:17) i + | j | z j ; (16)( ϕ | X )( z ) = (cid:88) j ϕ , j ( z ) ( − λ ) j . (17)(2) strictly meromorphic Jacobi forms ψ i, j satisfying ϕ ( z ) = (cid:88) i, j ψ i, j ( z ) e i A ( z ) j · · · A ( z n ) j n . (18) Proof.
The first part follows directly from the definition of quasi-Jacobi forms using thealgebraic independence of ν and ξ over the field of meromorphic functions, where ϕ i, j aregiven by the meromorphic almost Jacobi form Φ corresponding to ϕ byΦ( τ, z ) = (cid:88) i, j ϕ i, j ( τ, z ) ν ( τ ) i ξ ( τ, z ) j · · · ξ ( τ, z n ) j n . ψ i, j by the expansion of the almost Jacobiform Φ corresponding to ϕ , i.e.Φ( τ, z ) = (cid:88) i, j ψ i, j ( τ, z ) (cid:16) e ( τ ) + ν ( τ )2 π i (cid:17) i (cid:89) r ( A ( τ, z r ) + ξ ( τ, z r )) j r . Note that e ( τ ) + ν ( τ )2 π i and A ( τ, z r ) + ξ ( τ, z r ) transform as Jacobi forms. Moreover, they arealgebraically independent over the space of all meromorphic functions. Hence, it follows thatthe coefficients ψ i, j are strictly meromorphic Jacobi forms. The constant term with respectto i and j , by definition equal to ϕ , is now easily seen to equal to right-hand side of (18). Remark . In fact, the proof implies that ϕ i, j are quasi-Jacobi forms related to the ψ i, j by (2 π i) i ϕ i, j = (cid:88) i (cid:48) , j (cid:48) ψ i + i (cid:48) , j + j (cid:48) (cid:18) i + i (cid:48) i (cid:19)(cid:18) j + j (cid:48) j (cid:19) e ( τ ) i (cid:48) (cid:89) r A ( τ, z r ) j (cid:48) r , where (cid:18) j + j (cid:48) j (cid:19) = (cid:89) r (cid:18) j r + j (cid:48) r j r (cid:19) . Also, note that, given a representation for ϕ as in (18) one has ϕ , = 12 π i ∂∂ e ϕ and ϕ ,e i ( z ) = ∂∂A ( z i ) ϕ ( z ) . (cid:52) By virtue of (18) quasi-Jacobi forms share the properties of Jacobi forms with respect tothe location of the poles:
Corollary 2.24.
The statement of Theorem 2.12 also holds when ϕ is a strictly meromorphic quasi -Jacobi form. Another corollary of Proposition 2.9 is the following. Denoting by J and (cid:101) J the space of allstrictly meromorphic Jacobi forms and strictly meromorphic quasi-Jacobi forms, respectively,with all poles at the lattice points L τ , we have the following representations. J = C [ E − e , E , e , Θ] , (cid:101) J = C [ E , E , E , e , e , Θ] . In particular, given the weight and index, these spaces are finite dimensional.
Corollary 2.25.
The space of all meromorphic quasi-Jacobi forms of weight k , index M , andwith all poles in a finite union of rational hyperplanes as in Theorem 1.8 is finite dimensional.Proof. This follows directly from the previous proposition as by Corollary 2.14 the numberof linearly independent ψ i, j is finite. 18 .5. Action of the Jacobi Lie algebra by derivations The operators D τ = π i ∂∂τ and D z i = π i ∂∂z i preserve the space of quasi-Jacobi forms. Theseoperators are part of a Lie algebra of operators acting on quasi-Jacobi forms by derivations,as we explain now.Following a suggestion by Zagier, we consider the notion of a g -algebra for any Lie algebra g given by the following definition. Definition 2.26.
Given a Lie algebra g , a g -algebra is an algebra A together with a Liehomomorphism g → Der( A ) , where Der( A ) denotes the Lie algebra of all derivations on A .Denote by d τ and d z i the operators on the space of quasi-Jacobi forms given by ϕ (cid:55)→ π i ϕ , and ϕ (cid:55)→ ϕ ,e i respectively (see Proposition 2.22 and Remark 2.23). Then the functions ϕ i, j are given by ϕ i, j = d iτ i ! d jz j ! ϕ ( d jz = d j z · · · d j n z n , j ! = j ! · · · j n !) , so that the transformation behaviour of ϕ is uniquely determined by the action of the op-erators d τ and d z i . This observation is important to understand the next section, where weinvestigate how the transformation behaviour of a quasi-Jacobi form determines the transfor-mation of its Taylor coefficients, and vice versa, by studying the action of d τ on these Taylorcoefficients.Moreover, writing ϕ as in (18) yields d τ ϕ = ∂∂ e ϕ, ( d z i ϕ )( z ) = ∂∂A ( z i ) ϕ ( z i ) . Denote by W and I ij the weight and index operators acting diagonally by multiplying with theweight k and Q ( e i , e j ) respectively, where Q is the index. Let j the Lie algebra of the Jacobigroup. By [OP19, Eqn. (12)] the Lie algebra of the Jacobi group acts by the beforementionedoperators on the space of quasi-Jacobi forms. Proposition 2.27.
The algebra of quasi-Jacobi forms is a j -algebra, i.e. the algebra of deriva-tions D τ , D z i , d τ , d z i , W and I ij is isomorphic to j and acts on the space of quasi-Jacobi forms.Remark . More concretely, the commutation relations of (i) the modular operators, (ii)the elliptic operators and (iii) their interactions are given by(i) [ d τ , D τ ] = W , [ W, D τ ] = 2 D τ , [ W, d τ ] = − d τ , (ii) [ d z i , D z j ] = 2 I i,j , [ I ij , D z i ] = 0 , [ I ij , d z i ] = 0 , (iii) [ d z i , D τ ] = D z i , [ d τ , D z i ] = d z i , [ W, D z i ] = D z i . The other commutators vanish. As the spaces of almost Jacobi forms and quasi-Jacobi formsare isomorphic, the same result holds for almost Jacobi forms when one replaces d τ by 2 π i ∂∂ν and d z by ∂∂ξ ( z ) . (cid:52) There is yet another equivalent definition of quasi-Jacobi forms as derivatives of Jacobiforms. This definition only applies in case the index is positive definite. As a non-example,no power of the quasi-Jacobi form e (which is in fact a quasimodular form) can be writtenin terms of derivatives of Jacobi forms. 19 roposition 2.29. Let ϕ be a quasi-Jacobi form of weight k and positive definite index M .Then, there exist unique Jacobi forms ψ d with d ∈ Z n +1 ≥ of weight k − d − d − . . . − d n andindex M such that ϕ = (cid:88) d D d τ D d z · · · D d n z n ψ d . Proof.
Choose an ordering on Z n +1 respecting the ordering on Z . Given a Jacobi form ϕ ,let ( i, j ) be maximal (with respect to this ordering) for which ϕ i, j in Proposition 2.22 existsand is non-zero. A direct check using the same proposition shows that ϕ minus a multiple of D j τ D j z j · · · D j n z jn ϕ i, j is a quasi-Jacobi form for which this maximal index is smaller. Here, bythe positive definiteness of Q this multiple is non-zero. A holomorphic Jacobi form has two important representations: the theta expansion and the
Taylor expansion . We generalize the Taylor expansion to strictly meromorphic quasi-Jacobiforms in such a way that the Taylor coefficients are quasimodular forms. Moreover, we givecriteria based on the coefficients in these representations for a meromorphic function to be aquasi-Jacobi form.Given a Jacobi form ϕ and X ∈ M ,n ( Q ), the Taylor coefficients of ( ϕ | X )( z ) around z = are quasimodular forms for the groupΓ X = { γ ∈ SL ( Z ) | γX − X ∈ M ,n ( Z ), ρ ( X − γX ) = ζ X,γX − X } , (19)where as usual X = (cid:0) λµ (cid:1) and γX = (cid:0) λ γ µ γ (cid:1) and ρ and ζ X,X (cid:48) are defined by (9). In contrast toJacobi forms, it is not true that the Taylor coefficients of quasi-Jacobi forms are quasimodular.Namely, as stated in the introduction, for X = (cid:0) λµ (cid:1) ∈ M ,n ( Q ) one has that (Θ (cid:48) | X | γ )( τ, z )equals—up to the multiplicative constant ρ ( X ) ρ ( − Xγ )—(Θ (cid:48) | γX )( τ, z ) + czcτ + d (Θ | γX )( τ, z ) + λ (Θ | γX )( τ, z ) − λcτ + d (Θ | γX )( τ, z ) , (20)for all γ ∈ Γ X . All but the last term − λcτ + d (Θ | X )( τ, z ) of (20) depend polynomially on ccτ + d ,so that the Taylor coefficients of (Θ (cid:48) | X )( τ, z ) at z = 0 are not transforming in accordancewith the quasimodular transformation formula. Note the this last term can be writtenas − λ (Θ | X | γ )( τ, z ), where it should be noted that the weight 0 in the slash operator isunusual, corresponding to Θ (cid:48) rather than to Θ. In conclusion, the functionΘ (cid:48) (cid:107) X := ρ ( − X ) (cid:0) Θ (cid:48) | X + λ Θ | X (cid:1) rather than Θ (cid:48) | X transforms as a quasi-Jacobi form of weight 0, i.e. for γ ∈ Γ X one has(Θ (cid:48) (cid:107) X | γ )( τ, z ) = (Θ (cid:48) (cid:107) X )( τ, z ) + czcτ + d (Θ | X )( τ, z ) . Therefore, when we define the Taylor coefficients of a function ϕ at X ∈ M ,n ( Q ) in the nextsection, we in fact take the usual Taylor coefficients of ( ϕ (cid:107) X )( τ, z ), defined in the definitionbelow, around z = . 20 efinition 2.30. Given M ∈ M n, ( Q ) and a family of functions ϕ , j : h × C n → C indexedby a finite subset of Z n ≥ (with ϕ := ϕ , ), define the double slash operator by ϕ (cid:107) M X = ρ ( − X ) (cid:88) j ( ϕ , j | M X ) λ j , where ρ is given by (9). Convention 2.31.
In case ϕ is a quasi-Jacobi form, in this definition we always take thefamily ϕ , j determined by the elliptic transformation (18). Proposition 2.32.
Given a family of functions ϕ i, j : h × C n → C indexed by a finite subsetof Z ≥ × Z n ≥ (with ϕ = ϕ , ) and X ∈ M ,n ( R ) , one has (i) If ϕ satisfies the quasimodular transformation (16) for Γ , then ( ϕ (cid:107) X | γ )( τ, z ) = (cid:88) i, j ( ϕ i, j (cid:107) γX )( τ, z ) (cid:16) ccτ + d (cid:17) i + | j | z j for all γ ∈ Γ . (ii) If ϕ satisfies the quasi-elliptic transformation (17) , then ϕ (cid:107) X (cid:107) X (cid:48) = ϕ (cid:107) X (cid:48) (cid:107) X = ρ ( X (cid:48) ) ϕ (cid:107) X, ϕ (cid:107) X + X (cid:48) = ζ X,X (cid:48) ϕ (cid:107) X (cid:107) X (cid:48) for all X (cid:48) ∈ M ,n ( Z ) , where the root of unity ζ X,X (cid:48) is defined by (9) . (iii) If ϕ is a quasi-Jacobi form for SL ( Z ) , then ϕ (cid:107) X is a quasi-Jacobi form for Γ X , and ϕ (cid:107) X + X (cid:48) = ρ ( X (cid:48) ) ζ X,X (cid:48) ϕ (cid:107) X for all X (cid:48) ∈ M ,n ( Z ) .Proof. The transformation of ϕ (cid:107) X under γ and X follows by direct computations. We oftenmake use of d i (cid:48) τ i (cid:48) ! d j (cid:48) z j (cid:48) ! ϕ i, j = d i (cid:48) τ i (cid:48) ! d j (cid:48) z j (cid:48) ! d iτ i ! d jz j ! ϕ = (cid:18) i + i (cid:48) i (cid:19) (cid:18) j + j (cid:48) j (cid:19) ϕ i + i (cid:48) , j + j (cid:48) , where (cid:18) j + j (cid:48) j (cid:19) = (cid:89) r (cid:18) j r + j (cid:48) r j r (cid:19) . For the first property: ϕ (cid:107) X | k γ = ρ ( − X ) (cid:88) (cid:96) ( ϕ , (cid:96) | X | k γ ) λ (cid:96) = ρ ( − γX ) (cid:88) (cid:96) ( ϕ , (cid:96) | k γ | γX ) λ (cid:96) = ρ ( − γX ) (cid:88) i, j , (cid:96) ( ϕ i, j + (cid:96) | γX ) (cid:16) ccτ + d (cid:17) i (cid:18) j + (cid:96)j (cid:19) (cid:16) c ( z + λ γ τ + µ γ ) cτ + d (cid:17) j (cid:16) λ cτ + d (cid:17) (cid:96) ρ ( − γX ) (cid:88) i, j ( ϕ i, j | γX ) (cid:16) ccτ + d (cid:17) i (cid:16) c ( z + λ γ τ + µ γ ) + λ cτ + d (cid:17) j = ρ ( − γX ) (cid:88) i, j ( ϕ i, j | γX ) (cid:16) ccτ + d (cid:17) i (cid:16) c z cτ + d + λ γ (cid:17) j = ρ ( − γX ) (cid:88) i, j , (cid:96) ( ϕ i, j + (cid:96) | γX ) (cid:16) ccτ + d (cid:17) i (cid:18) j + (cid:96)j (cid:19) (cid:16) c z cτ + d (cid:17) j ( λ γ ) (cid:96) = (cid:88) i, j ( ϕ i, j (cid:107) γX ) (cid:16) ccτ + d (cid:17) i (cid:16) c z cτ + d (cid:17) j . For the second property, observe that ϕ (cid:107) X (cid:107) X (cid:48) = ρ ( X (cid:48) ) (cid:88) (cid:96) ( ϕ , (cid:96) (cid:107) X | X (cid:48) ) ( λ (cid:48) ) (cid:96) = ρ ( X ) ρ ( X (cid:48) ) (cid:88) j , (cid:96) ( ϕ , j + (cid:96) | X | X (cid:48) ) (cid:18) j + (cid:96)j (cid:19) λ j ( λ (cid:48) ) (cid:96) , from which it is clear that ϕ (cid:107) X (cid:107) X (cid:48) = ϕ (cid:107) X (cid:48) (cid:107) X . Moreover, by using the elliptic transforma-tion, we find it equals ρ ( X ) ρ ( X (cid:48) ) (cid:88) j , (cid:96) , m ( ϕ , j + (cid:96) + m | X ) (cid:18) j + (cid:96) + mj , (cid:96) , m (cid:19) λ j ( λ (cid:48) ) (cid:96) ( − λ (cid:48) ) m = ρ ( X ) ρ ( X (cid:48) ) (cid:88) j ( ϕ , j | X ) λ j = ρ ( X (cid:48) ) ϕ (cid:107) X. Next, one has( ϕ (cid:107) X + X (cid:48) ) = ρ ( X + X (cid:48) ) (cid:88) (cid:96) ( ϕ , (cid:96) | X + X (cid:48) ) ( λ + λ (cid:48) ) (cid:96) = ρ ( X ) ρ ( X (cid:48) ) ζ X,X (cid:48) (cid:88) (cid:96) ( ϕ , (cid:96) | X | X (cid:48) ) ( λ + λ (cid:48) ) (cid:96) = ρ ( X ) ρ ( X (cid:48) ) ζ X,X (cid:48) (cid:88) j , (cid:96) ( ϕ , j + (cid:96) | X ) (cid:18) j + lj (cid:19) ( − λ (cid:48) ) j ( λ + λ (cid:48) ) (cid:96) = ρ ( X ) ρ ( X (cid:48) ) ζ X,X (cid:48) (cid:88) j ( ϕ , j | X ) λ j = ρ ( X (cid:48) ) ζ X,X (cid:48) ϕ (cid:107) X. Finally, the fact that ϕ (cid:107) X is a quasi-Jacobi form follows directly from the definition of Γ X and the previous properties. Let X = (cid:0) λµ (cid:1) ∈ M ,n ( Q ) , M ∈ M ,n ( Q ) and ϕ ∈ Mer . We now study the Taylor coefficientsof ϕ (cid:107) M X around z = . In case ϕ is a strictly meromorphic quasi-Jacobi form, recall that22ll poles z lie on a hyperplane of the form s · z ∈ pτ + q + L τ for some s ∈ Z n and p, q ∈ Q / Z by Theorem 2.12ii. From now on we assume that s = e i for some i , so that a Laurent seriesof ϕ (cid:107) X of the form (cid:88) (cid:96) ≥ L · · · (cid:88) (cid:96) n ≥ L a (cid:96) ,...,(cid:96) n ( z − λ τ − µ ) (cid:96) · · · ( z n − λ n τ − µ n ) (cid:96) n for some L ∈ Z and a (cid:96) ∈ C exists. For example, the poles of all the meromorphic quasi-Jacobiforms we encounter in the applications lie on the coordinate axes. Definition 2.33.
We call the poles of a meromorphic function ϕ : h × C n → C orthogonal ifthe set of poles of ϕ ( τ, · ) is given by a union of special hyperplanes of the form z s ∈ pτ + q + L τ for some s ∈ { , . . . , n } and p, q ∈ Q / Z .Now that we introduced orthogonal poles, above, and the double slash action in Defini-tion 2.30, we defined the “Taylor coefficients” of a family of functions in the following way.Recall that in case ϕ is a Jacobi form, there is a canonical choice for the family of functions ϕ which is part of the data of these “Taylor coefficients” (see Convention 2.31). Definition 2.34.
Let M ∈ M n, ( Q ) and ϕ = { ϕ i, j } , where ϕ i, j : h × C n → C , be a familyof meromorphic functions indexed by a finite subset of Z ≥ × Z n ≥ , with ϕ := ϕ , ∈ Mer M such that all poles of ϕ are orthogonal. Let g (cid:96) ( ϕ ) as the (cid:96) th Laurent coefficient of ϕ :( ϕ )( τ, z ) = (cid:88) (cid:96) g (cid:96) ( ϕ )( τ ) z (cid:96) . For all X ∈ M ,n ( R ), we define the “Taylor coefficient” g X (cid:96) ( ϕ ) as g (cid:96) ( ϕ (cid:107) M X ). Also, denote g X (cid:96) ,s ( ϕ ) = g (cid:96) (cid:16)(cid:88) i + | j | = s ( ϕ i, j (cid:107) X )( z ) z j (cid:17) ( s ∈ Z ≥ ) . Remark . One may be tempted to write“ g X (cid:96) ,s ( ϕ ) = (cid:88) i + | j | = s g (cid:96) − j ( ϕ i, j (cid:107) X ) ” . However, we do not assume that the functions ϕ i, j admit orthogonal poles, so the Taylorexpansion of ϕ i, j may not exist. For example, taking ϕ = F (defined by Definition 1.7),we will see later that ϕ ,e ( z , z ) = z + z ) , which has a pole whenever z + z = 0.Theorem 2.41 implicitly shows that the notation g X (cid:96) ,s ( ϕ ) is well-defined for a quasi-Jacobiform ϕ with all poles orthogonal. (cid:52) The data { g X (cid:96) ( ϕ ) } uniquely determines ϕ as well as the family ϕ = { ϕ i, j } . Hence, itis natural to ask under which conditions on g X (cid:96) ( ϕ ) the function ϕ is a meromorphic quasi-Jacobi form. Before we answer this question, we study the modular properties of g X (cid:96) ( ϕ )given ϕ is a meromorphic quasi-Jacobi form. As a corollary of the previous propositionon ϕ (cid:107) X , generalizing [EZ85, Theorem 1.3] to quasi-Jacobi forms, we show that g X ( ϕ ) is aquasimodular form. 23 orollary 2.36. Let ϕ be a holomorphic quasi-Jacobi form of weight k and index M . Forall X ∈ M ,n ( Q ) , the function g X ( ϕ ) is a holomorphic quasimodular form of weight k for thegroup Γ X ( defined by (19)) . Moreover, d τ g X ( ϕ ) = g X ( d τ ϕ ) . Proof.
Let X ∈ M ,n ( Q ) and γ ∈ Γ X . Then, by Proposition 2.32iii one finds( g X ( ϕ ) | γ )( τ ) = ( ϕ (cid:107) X | γ )( τ, ) = (cid:88) i g X ( ϕ i ) (cid:16) ccτ + d (cid:17) i , where ϕ i denotes the family corresponding to ϕ i, . Hence, g X ( ϕ ) is a quasimodular formfor this group and d rτ g X ( ϕ ) = g X ( ϕ r r ! = g X ( d rτ ϕ ) . Holomorphicity in h and at infinity followsdirectly as ϕ is a holomorphic Jacobi form.The quasimodularity of the other coefficients g X (cid:96) ( ϕ ) can be understood in terms of lowercoefficients in two ways. First of all, certain linear combinations of derivatives of thesecoefficients are modular. Secondly the action of d iτ on g X (cid:96) ( ϕ ) can be expressed in terms ofother coefficients.We first show that these two ways are equivalent. Denote by ( x ) n the Pochhammer symbol( x ) n = x ( x + 1) · · · ( x + n − Proposition 2.37.
Let g = g k , g k − . . . , g k − p be quasimodular forms of depth at most p andweight k, k − . . . , k − p respectively. Then the following are equivalent: (i) d iτ g = g k − i for i = 0 , . . . , p ; (ii) The functions (cid:88) ≤ m ≤ p − i ( − m D m g k − i − m ( k − i − m − m m ! if k − p > or i < p − g − e g if k − p = 0 and i = p − for i = 0 , . . . , p − are modular forms of weight k − i ; (iii) The functions (cid:88) ≤ m ≤ p − i ( − m ( D + e ) m g k − i − m ( k − i − m − ) m m ! for i = 0 , . . . , p − are modular forms of weight k − i .Remark . Observe that as m ≤ p − i and i ≤ p − k − i − m − ≥ k − p. Hence, the numerator ( k − i − m − m vanishes in case k − p = 0 and i = p − m − g k − i − m is a modular form of weight 0, hence a constant function. Therefore,also the numerator D m g k − i − m vanishes in this case. One can think of e as being theappropriate regularization of this ill-defined ratio. Alternatively, in the third equivalence,one can replace D by D + e in which case the numerator never vanishes. Finally, if onewould replace D by e , one would obtain a generalisation of the functions ϕ n of [Bri18,Proposition 3.1]. (cid:52) roof. Assume that d iτ g = g i . As g is of depth at most p , this implies that d τ g i = g i − forall i , where g i is taken to be zero when it is not defined. Hence, using [ d τ , D τ ] = W (seeRemark 2.28), it follows that applying d τ to a term in the sum in (ii) yields( − m D mτ g k − i − m − ( k − i − m − m m ! − ( − m − D m − τ g k − i − m ( k − i − ( m − − m − ( m − , where the second term is taken to be zero when m = 0. Also the first term vanishes when m = p − i as f k − p − is set to be zero. Hence, after applying d τ the sum becomes a telescopingsum, equal to zero. Also d τ ( g − e g ) = g − g = 0. The third statements follows by thesame argument, mutatis mutandis.The converse statement follows inductively by using that all but two terms in the samesum equal to zero cancel, hence these terms are equal.In order to express the derivatives with respect to d τ of the coefficients of ϕ , we introducethe following notation describing how the index mixes the coefficients. Definition 2.39.
Given M ∈ M n, ( Q ), ϕ ∈ Mer M and a family ϕ as before, we write Q g X (cid:96) ,s ( ϕ )for Q r g X (cid:96) ,s ( ϕ ) = g (cid:96) ( Q rM ϕ (cid:107) M X ) ( r, s ∈ Z ≥ , l ∈ Z n , X ∈ M ,n ( Q )) . Moreover, the functions ξ X (cid:96) ( ϕ )( τ ) are defined by (again l ∈ Z n , X ∈ M ,n ( Q )) ξ X (cid:96) ( ϕ ) := (cid:88) r (cid:88) s ≤ r ( − r D rτ ( Q r − s g X (cid:96) ,s )( ϕ )( k + | (cid:96) | − s − r ( r − s )! g X (cid:96) ( ϕ ) − e (( Q g X (cid:96) )( ϕ ) + ( g X (cid:96) , )( ϕ )) k = 0 , | (cid:96) | = 2 , (21)where ( x ) n = x ( x + 1) · · · ( x + n −
1) denotes the Pochhammer symbol. Abbreviate ξ X (cid:96) ( ϕ )by ξ (cid:96) ( ϕ ) if X is the zero matrix. Remark . We formulate all results below for ξ X (cid:96) ( ϕ ) as defined above, but by Proposi-tion 2.37 all results remain valid after replacing ξ X (cid:96) ( ϕ ) by (cid:88) r ( − r (cid:88) s ( D τ + e ) r ( Q r − s g X (cid:96) ,s )( ϕ )( k + | (cid:96) | − r − ) r ( r − s )! . Note that this equation, as well as Equation 21, can be inverted, expressing ξ X (cid:96) ( ϕ ) as linearcombination of derivatives of certain g X m ( ϕ ). (cid:52) This allows to characterize invariance under the modular action by its Taylor coefficients,generalizing [EZ85, Theorem 3.2]:
Theorem 2.41.
Let Γ be a congruence subgroup, k ∈ Z and M ∈ M n ( Q ) . Let ϕ = { ϕ i, j } asbefore a family of meromorphic functions h × C n → C with ϕ := ϕ , ∈ Mer M and admittinga Laurent expansion around z = . Then the following are equivalent: The function ϕ satisfies the quasimodular transformation (16)( ϕ | k,Q γ )( τ, z ) = (cid:88) i, j ϕ i, j ( τ, z ) (cid:16) ccτ + d (cid:17) i + | j | z j for all γ ∈ Γ ; (ii) The coefficients g (cid:96) ( ϕ ) are quasimodular forms of weight k + | (cid:96) | uniquely determined bythe coefficients of the ϕ i, j , i.e. d rτ g (cid:96) ( ϕ ) = (cid:88) s r !( r − s )! Q r − s g (cid:96) ,s ( ϕ ) ;(iii) The functions ξ (cid:96) ( ϕ ) are modular forms of weight k + | (cid:96) | on Γ for all (cid:96) .Proof. Expanding (16) yields (cid:88) (cid:96) g (cid:96) ( ϕ )( γτ )( cτ + d ) k +2 m + s z (cid:96) = sum (cid:96) (cid:88) r,s g (cid:96) ,s ( ϕ )( τ ) r ! (cid:16) ccτ + d (cid:17) r + s Q M ( z ) r z (cid:96) Extracting on both sides the coefficient of z (cid:96) yields d rτ g (cid:96) ( ϕ ) = (cid:88) s r !( r − d )! Q r − s g (cid:96) ,s ( ϕ ) . Also, the coefficient g (cid:96) ( ϕ ) is holomorphic in h as well as at the cusps of Γ, because of theanalytic properties of the functions ϕ i, j . The rest of the statement follows from Proposi-tion 2.37.Using this result, one can characterize a quasi-Jacobi form ϕ by its Taylor coefficients inthree ways, i.e. by considering g X (cid:96) ( ϕ ) as a vector-valued quasimodular form, by the modularityof the functions ξ X (cid:96) ( ϕ ), and finally by the action of d τ on the quasimodular form g X (cid:96) ( ϕ ).Writing f X for g X (cid:96) ( ϕ ) or ξ X (cid:96) ( ϕ ), the ‘elliptic transformation’ of the quasimodular form f X is given by f X + X (cid:48) = ρ ( X (cid:48) ) ζ X,X (cid:48) f X for all X (cid:48) ∈ M ,n ( Z ) (22)Recall that g X (cid:96) ( ϕ ) and ξ X (cid:96) ( ϕ ) are only defined when the zeros of ϕ are orthogonal (see Defini-tion 2.33) and depend on a family of functions ϕ i, j : h × C n → C . In case ϕ is a quasi-Jacobiform this family ϕ = { ϕ i, j } determines the transformation of ϕ ; without the assumptionthat ϕ is as a quasi-Jacobi form we have the following statement refining Theorem 1.9. Theorem 2.42.
Let k ∈ Z , M ∈ M ,n ( Q ) and ϕ ∈ Mer Mn such that the set of poles of ϕ isorthogonal. Given a family ϕ = { ϕ i, j } , indexed by ( i, j ) in a finite subset of Z ≥ × Z n ≥ , ofmeromorphic functions ϕ i, j : h × C n → C with ϕ = ϕ , , the following are equivalent: (i) The function ϕ is a meromorphic quasi-Jacobi form of weight k and index M for whichthe functions ϕ i, j determine its transformation behaviour. For all X = (cid:0) λµ (cid:1) ∈ M ,n ( Q ) with λ τ + µ not a pole of ϕ , the function g X ( ϕ ) is a vectorvalued quasimodular form satisfying (22) and transforming as g X ( ϕ ) | k γ = (cid:88) s g γX ,s ( ϕ ) (cid:16) ccτ + d (cid:17) s . (ii (cid:48) ) For all X ∈ M ,n ( Q ) the function g X (cid:96) ( ϕ ) is a vector valued quasimodular form satisfy-ing (22) and transforming as g X (cid:96) ( ϕ ) | k γ = (cid:88) r (cid:88) s r − s )! Q r − s g γX (cid:96) ,s ( ϕ ) (cid:16) ccτ + d (cid:17) r . (iii) For all (cid:96) ∈ Z n the functions ξ (cid:96) ( ϕ ) are modular forms of weight k + | (cid:96) | for SL ( Z ) andfor all X ∈ M ,n ( Q ) the functions ξ X ( ϕ ) satisfy (22) . (iii (cid:48) ) For all X ∈ M ,n ( Q ) and (cid:96) ∈ Z n the functions ξ X (cid:96) ( ϕ ) in (21) are modular forms ofweight k + | (cid:96) | for Γ X and satisfy (22) . (iv) For all X ∈ M ,n ( Q ) and (cid:96) ∈ Z n the functions g X (cid:96) ( ϕ ) are quasimodular forms ofweight k + | (cid:96) | for Γ X , satisfying (22) and d rτ g X (cid:96) ( ϕ ) = (cid:88) s r !( r − s )! ( Q r − s g X (cid:96) ,s )( ϕ ) . (23) Proof. (i) implies (iv): Let X ∈ M ,n ( Q ) and γ ∈ Γ X be given. By Proposition 2.32 thefunction ϕ (cid:107) X satisfies the conditions of Theorem 2.41 for Γ = Γ X . Moreover, by the sameproposition the coefficients only depend on X mod M ,n ( Z ) up to the factor ρ ( X (cid:48) ) ζ X,X (cid:48) .(iv) implies (iii (cid:48) ): This follows directly from Theorem 2.41 for Γ = Γ X .(iii (cid:48) ) implies (iii): Observe that Γ X = SL ( Z ) for X equal to the zero matrix. Hence, wesimply forget some of the properties of ξ X (cid:96) .(iii) implies (ii) (cid:48) : As ξ (cid:96) is a modular form for SL ( Z ) for all (cid:96) ∈ Z n , it follows by Theo-rem 2.41 that ϕ satisfies the quasimodular transformation for all γ ∈ SL ( Z ). As by Propo-sition 2.32 ϕ (cid:107) X | k γ = (cid:88) i, j ( ϕ i, j (cid:107) γX ) (cid:16) ccτ + d (cid:17) i (cid:16) c z cτ + d (cid:17) j , the result follows by extracting the coefficients of z on both sides. Finally, note that ξ X = g X .(ii (cid:48) ) implies (ii): This follows directly by restricting to (cid:96) = .(ii) implies (i): Suppose z = λ τ + µ with X = (cid:0) λµ (cid:1) ∈ M ,n ( Q ) not being a pole of ϕ .Let ϕ i be the family of functions corresponding to ϕ i, . Using (ii), for γ ∈ SL ( Z ) one hasthat( g X ( ϕ )) | k γ = (cid:88) i g γX ( ϕ i )( τ ) (cid:16) ccτ + d (cid:17) i = ρ ( − γX ) (cid:88) i, (cid:96) ( ϕ i, (cid:96) | γX )( τ, (cid:16) ccτ + d (cid:17) i ( λ γ ) (cid:96) ρ ( − γX ) (cid:88) i, j , (cid:96) ( ϕ i, j + (cid:96) | γX )( τ, (cid:16) ccτ + d (cid:17) i (cid:18) j + (cid:96)j (cid:19)(cid:16) c ( λ γ τ + µ γ ) cτ + d (cid:17) j λ (cid:96) . On the other hand, ( g X ( ϕ )) | k γ = ρ ( − X ) (cid:88) (cid:96) ( ϕ , l | X | k γ )( τ, λ (cid:96) = ρ ( − γX ) (cid:88) (cid:96) ( ϕ , l | k γ | γX )( τ, λ (cid:96) . Combining the identities yields (cid:88) (cid:96) ( ϕ , l | k γ | Xγ )( τ, λ (cid:96) = (cid:88) i, j , (cid:96) ( ϕ i, j + (cid:96) | γX )( τ, (cid:16) ccτ + d (cid:17) i (cid:18) j + (cid:96)j (cid:19)(cid:16) c ( λ γ τ + µ γ ) cτ + d (cid:17) j λ (cid:96) . As both sides equal ρ ( γX ) g X ( ϕ ), which is periodic with finite period as a function of λ , theconstant terms with respect to λ agree. Hence,( ϕ | γ | X )( τ,
0) = (cid:88) i, j ( ϕ i, j | X )( τ, (cid:16) ccτ + d (cid:17) i (cid:16) c ( λ τ + µ ) cτ + d (cid:17) j for all X = (cid:0) λµ (cid:1) ∈ M ,n ( Q ) with X not corresponding to a pole. Therefore, ϕ satisfies thequasimodular transformation for all z of the given form. As ( Q n τ × Q ) n \ P ϕ , with P ϕ theset of poles of ϕ , lies dense in C n for all τ ∈ h , the function ϕ satisfies the quasimodulartransformation equation.For the elliptic transformation, we again assume z = λ τ + µ with X = (cid:0) λµ (cid:1) ∈ M ,n ( Q )not being a pole. Given X (cid:48) = ( λ (cid:48) , µ (cid:48) ) ∈ M ,n ( Z ), we have (cid:88) l ( ϕ , l | X )( τ, λ (cid:96) = ρ ( X ) g X ( ϕ )= ρ ( X ) ρ ( − X (cid:48) ) ζ − X + X (cid:48) g X + X (cid:48) ( ϕ )= ρ ( − X (cid:48) ) ζ − X + X (cid:48) (cid:88) l ( ϕ , l | X (cid:48) | X )( τ,
0) ( λ + λ (cid:48) ) (cid:96) . The coefficients of λ agree, so( ϕ | X (cid:48) )( τ, z ) = ρ ( X (cid:48) ) ζ X + X (cid:48) (cid:88) j ϕ , j ( τ, z ) ( − λ (cid:48) ) j for all z of the given form. As before by continuity of ϕ the above equation holds for all z . Remark . The proof of the above result also applies to weak
Jacobi forms, after re-placing ‘meromorphic Jacobi form’ and ‘(quasi)modular’ by ‘weak Jacobi form’ and ‘weaklyholomorphic (quasi)modular’, respectively. (cid:52)
Specializing to holomorphic Jacobi forms (instead of meromorphic quasi -Jacobi forms),we obtain the following result, generalizing the main results on Taylor coefficients of Jacobiforms in [EZ85] to multivariable Jacobi forms.28 orollary 2.44.
Let k ∈ Z , M ∈ M ,n ( Q ) and ϕ ∈ Hol Mn . Then, the following are equivalent: (i) The function ϕ is a holomorphic Jacobi form of weight k and index M . (ii) For all X ∈ M ,n ( Q ) the function g X ( ϕ ) is a vector valued modular form satisfying (22) and transforming as g X ( ϕ ) | k γ = g γX ( ϕ ) . (ii (cid:48) ) For all X ∈ M ,n ( Q ) the function g X (cid:96) ( ϕ ) is a vector valued quasimodular form satisfy-ing (22) and transforming as g X (cid:96) ( ϕ ) | k γ = (cid:88) r r ! ( Q r g γX (cid:96) )( ϕ ) (cid:16) ccτ + d (cid:17) r . (iii (cid:48) ) For all X ∈ M ,n ( Q ) and (cid:96) ∈ Z n the functions ξ X (cid:96) ( ϕ ) given by ξ X (cid:96) ( ϕ ) = (cid:88) r ( − r D rτ ( Q r g X (cid:96) )( ϕ )( k + | (cid:96) | − r r ! are modular forms of weight k + | (cid:96) | for SL ( Z ) and satisfy (22) . (iv) For all X ∈ M ,n ( Q ) and (cid:96) ∈ Z n the functions g X (cid:96) ( ϕ ) are quasimodular forms ofweight k + | (cid:96) | for Γ X , satisfying (22) and d rτ g X (cid:96) ( ϕ ) = ( Q r g X (cid:96) )( ϕ ) .
3. Quasimodular algebras for congruence subgroups
The main result of the previous section, Theorem 2.42, will imply almost immediately theproof of Theorem 1.1 on the quasimodularity of the elements of Λ ∗ ( N ) for some subgroup. Wewill first introduce a more general set-up, in the context of which we will present this proof.In the rest of this section we provide many examples of algebras of functions on partitions towhich this general set-up applies, i.e. we will recall the hook-length moments and the (double)moments functions and explain how the corresponding algebras can be extended to severalcongruence subgroups. We answer the question how to extend an algebra of functions on partitions for which the q -bracket is a quasimodular form for SL ( Z ) to one for a congruence subgroup Γ. More precisely,we consider quasimodular algebras for Γ. Definition 3.1. A quasimodular algebra for a congruence subgroup Γ ≤ SL ( Z ) is a gradedalgebra of functions f on the set of partitions for which (cid:104) f (cid:105) q is a quasimodular form for Γ ofthe same weight as f . 29e now present a construction of a quasimodular algebra for a congruence subgroup givena quasimodular algebra for SL ( Z ). In order to do so, from now on we assume that Φ : P × C r → C is such that for all n ≥ k ∈ Z such that the function ϕ Φ n : h × M n,r ( C ) → C given by ϕ Φ n ( τ, Z ) := (cid:68) n (cid:89) i =1 Φ( · , Z i ) (cid:69) q , where Z i is the i th row of Z , is a meromorphic quasi-Jacobi form of weight kn which ad-mits a Laurent expansion around all Z ∈ M n,r ( Q ) (after identifying M n,r ( C ) with C nr ).Here, Φ( λ, z ) can be thought of a generalisation of the generating series W λ ( z ) of the Bloch–Okounkov functions Q k , defined by (2). Definition 3.2.
Given such a Φ, for a ∈ Q r denote by f Φ (cid:96) ( · , a ) = f (cid:96) ( · , a ) : P → C the (cid:96) thTaylor coefficient of Φ( z ) around z = a , i.e.Φ( · , z ) = (cid:88) (cid:96) f (cid:96) ( · , a ) ( z − a ) (cid:96) . Define the graded Q -algebra F Φ ( N ) = F ( N ) as the algebra generated by the weight k + | (cid:96) | elements f (cid:96) ( · , a ) for a ∈ N Z r , (cid:96) ∈ Z r . Remark . By Theorem 2.42, up to a sign f m ( · , a ) and f m ( · , b ) agree whenever a − b ∈ Z r .Hence, in the definition one can assume that a ∈ [0 , r . (cid:52) For example, Q (cid:96) +1 ( a ) = e ( − a ) f W(cid:96) ( a ) (see (3)) and Λ ∗ ( N ) = F W ( N ).Let L ∈ M n,r ( Z ) and A ∈ M n,r ( Q ). An arbitrary monomial f L in F Φ ( N ) is given by f L ( A ) := f L ( A ) · · · f L n ( A n ) (24)with L i and A i the i th row of L and A , respectively. By construction of the Taylor coefficientsof Φ as well as of ϕ Φ n we find (cid:104) f Φ L ( A ) (cid:105) q = g (0 ,A ) L ( ϕ Φ n ) , where on the right-hand side we identified M n,r ( Z ) and M n,r ( Q ) with Z nr and Q nr , respec-tively, and as ϕ Φ n is a quasi-Jacobi form it uniquely determines the family of functions in thedefinition of the “Taylor coefficients” .The following result is the general statement of Theorem 1.1. Theorem 3.4.
Given Φ as above and N ≥ , let M = (2 , N ) N . The algebra F Φ ( N ) is aquasimodular algebra for w M Γ( M ) w − M .Proof. Consider a monomial element f L ( A ) of F ( N ) as in (24), for some L ∈ M n,r ( Z ) and A ∈ M n,r ( Q ).Write X = (0 , A ) t . Then, (cid:104) f L ( A ) (cid:105) q = g XL ( ϕ n ). This Taylor coefficient is quasimodularfor Γ X by Theorem 2.42. Therefore, it suffices to show that the q -bracket respects the weightgrading of F ( N ) and that Γ X contains w N Γ( N ) w − N . In fact, for X = (0 , A ) the double slash operator (cid:107) X coincides with the slash operator | X , so that the“Taylor coefficients” do not involve this family. f is given by (cid:80) ni =1 ( k + | L i | ), whereas correspond-ingly the weight of g X m ( ϕ ) equals kn + | L | (here | L | = (cid:80) i,j L ij ).Write M for the index of ϕ Φ n and Q = Q M for corresponding quadratic form. RecallΓ X = { γ ∈ SL ( Z ) | Xγ − X ∈ M n, ( Z ), ρ ( X − Xγ ) = ζ X,Xγ − X } . Writing γ = (cid:0) a bc d (cid:1) , we have Xγ − X = ( c a , ( d − a ) , ρ ( Xγ − X ) = e (( c − c ( d −
1) + ( d − ) Q ( a ))and ζ X,Xγ − X = e ( Q ( , ( d − a ) − Q ( c a , a )) = e ( − c Q ( a )) . Observe that 2 N Q ( a ) is integral. Hence, if γ ∈ SL ( Z ) satisfies c ≡ N, d ≡ N and c − c ( d −
1) + ( d − ≡ c mod 2 N , (25)then γ ∈ Γ X .Let N (cid:48) ∈ Z > . Then, w N (cid:48) Γ( N (cid:48) ) w − N (cid:48) = { (cid:0) a bc d (cid:1) ∈ SL ( Z ) | c ≡ N (cid:48) ) , a ≡ d ≡ N (cid:48) ) } . In case 2 (cid:45) N , the conditions (25) are satisfied for all γ ∈ w N (cid:48) Γ( N (cid:48) ) w − N (cid:48) when N (cid:48) = N , incase 2 | N for N (cid:48) = 2 N . Therefore, indeed, w M Γ( M ) w − M ≤ Γ X .For a monomial element f L ( A ) as in (24) with L ∈ M n,r ( Z ) and A ∈ M n,r ( Q ), and γ ∈ Γ ( N ) one has that (cid:104) f L ( A ) (cid:105) q (cid:12)(cid:12) γ = e (( c − cd + ( d − ) Q ( a )) (cid:104) f L ( A ) (cid:105) q , (26)where Q is again the (quadratic form corresponding to the) index of ϕ Φ n . Hence, restrictingto A ∈ M n,r ( Q ) (cid:39) Q nr for which Q ( A ) ∈ Z we find the following result, from which we areable to derive Theorem 1.3 using some additional properties of the Bloch–Okounkov n -pointfunctions. Proposition 3.5.
Given N ≥ , for all L ∈ M n,r ( Z ) and A ∈ N M n,r ( Z ) satisfying Q ( A ) ∈ Z + N Z ,one has that (cid:104) f L ( A ) (cid:105) q is a quasimodular form for Γ ( N ) .Proof. This follows from the observation (26) using the following two observations. First ofall, c − cd + ( d − ≡ N when c ≡ , d ≡ N . Secondly, for integers c, d theinteger c − cd + ( d − is always even whenever not both c and d are even.The functions Q ( p ) k in [GJT16] are given by Q ( p ) k ( λ ) = β ( p ) k + (cid:88) gcd(2 λ i − i +1 ,p )=1 (cid:0) ( λ i − i + ) k − − ( − i + ) k − (cid:1) , where β ( p ) k = β k (0)(1 − p ). Observe that Q ( p ) k ( λ ) = p − p − (cid:88) a =0 e ( a/p ) Q k (2 a/p )( λ ) . (27)31utside of the context of p -adic modular forms, there is no need to restrict p to be a prime,so (27) serves as the definition of Q ( m ) k for an integer m . Similarly, we define functions f ( d ) (cid:96) in terms of the Taylor coefficients f (cid:96) ( a ), as follows. Definition 3.6.
Let Φ be as above. Given d ∈ Z r> , we let U ( d ) = (cid:8) , d , . . . , d − d (cid:9) × · · · × (cid:8) , d r , . . . , d r − d r (cid:9) and for (cid:96) ∈ Z r define f ( d ) , Φ (cid:96) ( λ ) := f ( d ) (cid:96) ( λ ) := (cid:88) a ∈ U ( d ) f Φ (cid:96) ( λ, a ) ( λ ∈ P ) . Define the graded algebra F ( N ) , Φ as the Q -algebra generated by the functions f ( d ) , Φ (cid:96) for all (cid:96) ∈ Z r and d ∈ Z r> for which d i | N for all i .Then, Theorem 1.5 follows directly from the following result. Theorem 3.7.
Let Φ be as above. Given N ≥ , the algebra F ( N ) , Φ is a quasimodular algebrafor the congruence subgroup Γ ( N ) .Proof. Consider the monomial elements f ( D ) L := f ( D ) L · · · f ( D n ) L n in F ( N ) , where L, D ∈ M nr ( Z ).Everywhere in this proof we identify M n,r ( Z ) with Z nr . Then, f L ( D ) = (cid:88) A ∈ U ( D ) f L (2 A ) . Now, by part (ii (cid:48) ) in Theorem 2.42, for all γ = (cid:0) a bc d (cid:1) ∈ SL ( Z ) one has that (cid:104) f L (2 A ) (cid:105) q | γ = (cid:88) r h r (2 cA, dA ) (cid:16) ccτ + d (cid:17) r , where h r ( X ) = (cid:80) s r − s )! Q r − s g X (cid:96) ,s ( ϕ n ). If γ ∈ Γ ( N ), then 2 cA ∈ N M n,r ( Z ). Hence, for X = (0 , dA ) t and X (cid:48) = (2 cA, t one has ρ ( X (cid:48) ) ζ X,X (cid:48) = 1. Therefore, h r (2 cA, dA ) = h r ( , dA ) and h r ( , dA + B ) = h r ( , dA )for all B ∈ M n,r ( Z ). As 2 dA ranges over the same values modulo 2 as 2 A does for A ∈ U ( D ),one finds (cid:104) f ( D ) L (cid:105) q | γ = (cid:88) r (cid:88) A ∈ U ( D ) h r ( , A ) (cid:16) ccτ + d (cid:17) r for all γ ∈ Γ ( N ). Hence, indeed, (cid:104) f ( D ) L (cid:105) q is a quasimodular form for Γ ( N ).The rest of this chapter is devoted to providing examples of quasimodular algebras of higherlevel, using Theorem 3.7. 32 .2. First application: the Bloch–Okounkov theorem of higher level The construction and results on the Bloch–Okounkov algebra, as stated in the introduction,are proven in this section. In fact, this proof is an almost immediately consequence of theresults in the previous section using the properties of the Bloch–Okounkov n -point functions.Recall the Bloch–Okounkov n -point functions F n defined in Definition 1.7 as follows. Forall n ≥
0, let S n be the symmetric group on n letters and let F n ( τ, z , . . . , z n ) = (cid:88) σ ∈ S n V n ( τ, z σ (1) , . . . , z σ ( n ) ) , where the functions V n are defined recursively by V ( τ ) = 1 and n (cid:88) m =0 ( − n − m ( n − m )! θ ( n − m ) ( τ, z + . . . + z m ) V m ( τ, z , . . . , z m ) = 0 . These n -point functions are quasi-Jacobi forms of which we determine the weight and index(or rather the quadratic form uniquely determining a symmetric matrix M ∈ M n, ( Q ) whichis the index). Lemma 3.8.
The n -point functions F n are meromorphic quasi-Jacobi forms of weight n andindex Q ( z ) = − ( z + . . . + z n ) .Proof. We start with the observation that for all n ≥ Θ ( n ) ( z )Θ( z ) is a true mero-morphic Jacobi form (of weight n and index Q ( z ) = 0), in contrast to Θ( z ) itself which is weakly holomorphic. Namely, all poles are given by z ∈ Z τ + Z and for z = a + bτ with a, b ∈ Q , one has that Θ ( n ) ( aτ + b )Θ( aτ + b ) = (cid:80) ν ∈ F ν n e ( νb ) q ν / aν (cid:80) ν ∈ F e ( νb ) q ν / aν → − a, whenever τ → ∞ , or equivalently q → z + . . . + z n ) V n ( z , . . . , z n ) can be written as a polynomial ofweight n − Θ ( i ) ( z + ... + z j )Θ( z + ... + z j ) for i, j = 1 , . . . , n ; a fact which can be proven inductivelyby its recursion (6). Hence, Θ( z + . . . + z n ) V n ( z , . . . , z n ) is a meromorphic Jacobi form.As Θ( z ) − is a meromorphic Jacobi form of weight 1 and index Q ( z, z ) = − z , we concludethat F n is a meromorphic quasi-Jacobi form of weight n and index Q ( z ) = − | z | .Observe that Theorem 1.1 is a direct corollary of the previous lemma and Theorem 3.4.Also, Theorem 1.5 follows directly from the previous lemma and Theorem 3.7. So, we areonly left with the following proof. Proof of Theorem 1.3.
First of all, in case a ∈ Z , one has Q ( a ) = − | a | ∈ Z . Hence, theresult follows directly from Proposition 3.5.In the second case, for both (cid:104) Q k ( a ) (cid:105) q and (cid:104) Q ( a ) (cid:105) q the root of unity in (26) is e ( Q ( a )) = e ( Q ( a )), so that again the subgroup of quasimodularity is Γ ( N ).For the holomorphicity in the second case, observe that (cid:104) Q ( a ) (cid:105) q equals Θ( a ) − up to aconstant. Also, (cid:104) Q k ( a ) (cid:105) q can be written as a product of Taylor coefficients of the func-tion Θ( z + . . . + z n + a ) − and of Θ( z + . . . + z n + a ) F n ( z + a , . . . , z n + a n ), the latter33aylor coefficients being holomorphic quasimodular forms. Observe that the Taylor coeffi-cients around z = . . . = z n = 0 of Θ( a )Θ( z + . . . + z n + a )are all polynomials in the holomorphic quasimodular forms Θ ( i ) ( a )Θ( a ) . Therefore, (cid:104) Q k ( a ) (cid:105) q (cid:104) Q ( a ) (cid:105) q is aholomorphic quasimodular form. As a second example, for k ≥ H k ( λ ) = − B k k + (cid:88) ξ ∈ Y λ h ( ξ ) k − , where Y λ denotes the Young diagram of λ , ξ is a cell in this Young diagram, and h ( ξ )denotes the hook-length of this cell. By [CMZ18, Theorem 13.5] one has that H k is (up toa constant) equal to the ( k − W ( z ) W ( − z ). In particular,any homogeneous polynomial in the H k admits a quasimodular q -bracket. The results inSection 3.1 now specialize to the following statements. (i). For a ∈ Q and k ∈ Z ≥ , let H k ( a ) := ˜ α k ( a ) + 12 (cid:88) ξ ∈ Y λ (cid:0) e ( a h ( ξ )) + ( − k e ( − a h ( ξ )) (cid:1) h ( ξ ) k − , where (cid:80) k ≥ ˜ α k ( a ) z k − ( k − = sinh(( z + 2 π i a ) / − . Denote by H ( N ) the graded algebragenerated by the H k ( a ) with k ≥ N a ∈ Z . The algebra H ( N ) is graded by assigningto H k ( a ) weight k . Let M = (2 , N ) N . Corollary 3.9.
The algebra H ( N ) is quasimodular of level M , after scaling τ by M . More concretely, for all homogeneous f ∈ H ( N ) of some weight k , the rescaled q -bracket (cid:104) f (cid:105) q M ,where q M = q /M , is a quasimodular form of weight k for Γ( M ). (ii). Given N ≥ k ∈ Z n and a ∈ N Z n , write H k ( a ) = H k ( a ) · · · H k n ( a n ). Corollary 3.10.
For a ∈ N Z n with | a | ∈ Z and k ∈ Z n ≥ the q -bracket (cid:104) H k ( a ) (cid:105) q is aquasimodular form of weight | k | for Γ ( N ) . (iii). For k, t ∈ Z (cid:54) =0 , let H ( t ) k := − B k k t k + (cid:88) ξ ∈ Y λ h ( ξ ) ≡ t h ( ξ ) k − , which (up to a constant) also occurs in [BOW20].Denote by H ( N ) the algebra generated by the H ( t ) k for which k is even and t | N . Thisalgebra is graded by assigning weight k to H ( t ) k .34 orollary 3.11. The algebra H ( N ) is quasimodular for Γ ( N ) . More concretely, for all homogeneous f ∈ H ( N ) of weight k , the q -bracket (cid:104) f (cid:105) q is a quasi-modular form of weight k for Γ ( N ). Next, we consider the moment functions in [Zag16] S k ( λ ) = − B k k + ∞ (cid:88) i =1 λ k − i ( k ≥ . The generating series S ( z ) = z + (cid:80) k ≥ S k z k − ( k − satisfies [Itt20b, Corollary 3.3.2] G n ( z ) := (cid:104) S ( z ) · · · S ( z n ) (cid:105) q = 12 n +1 (cid:88) α ∈ Π( n ) (cid:89) A ∈ α (cid:88) s ∈{− , } | A | D | A |− τ E ( s · z A ) , where Π( n ) denotes the set of all set partitions of the set { , . . . , n } , | A | the cardinality ofthe set A , and z A = ( z a , . . . , z a r ) if A = { a , . . . , a r } . Hence, the n -point functions G n arequasi-Jacobi forms of weight 2 n and index zero. Because the index of G n is zero, the rootof unity in (26) vanishes for all γ ∈ SL ( Z ). Therefore, the results of Section 3.1 specializeto the following results for the groups Γ ( N ) and Γ ( N ) (rather than Γ( N ) and Γ ( N )respectively). (i). For a ∈ Q and k ≥
1, let S k ( a ) := α k ( a ) + 12 ∞ (cid:88) i =1 (cid:0) e ( aλ i ) + ( − k e ( − aλ i ) (cid:1) λ k − i , where (cid:80) k ≥ α k ( a ) z k − ( k − = ( e ( z + 2 π i a ) − − (these values agree with the constants ˜ α k inthe previous example, except if k = 1). Denote by S ( N ) the algebra generated by the S k ( a )with N a ∈ Z . Assign to S k ( a ) weight k . Corollary 3.12.
The algebra S ( N ) is quasimodular for Γ ( N ) . More concretely, for all homogeneous f ∈ H ( N ) of weight k , the q -bracket (cid:104) f (cid:105) q is a quasi-modular form of weight k for Γ ( N ). (ii). For k, t ∈ Z > , let S ( t ) k := − B k k t k + (cid:88) i ≥ λ i ≡ t λ k − i . Denote by S ( N ) the algebra generated by the S ( t ) k for which k is even and t | N . Assign to S ( t ) k weight k . Corollary 3.13.
The algebra S ( N ) is quasimodular for Γ ( N ) . More concretely, for all homogeneous f ∈ S ( N ) of weight k , the q -bracket (cid:104) f (cid:105) q is a quasi-modular form of weight k for Γ ( N ). 35 .5. Fourth application: double moment functions of higher level As a final example, consider the double moment functions introduced in [Itt20b] for twointeger parameters k ≥ , l ≥ T k,l ( λ ) = C k,l + ∞ (cid:88) m =1 m k F l ( r m ( λ )) . Here, C k,l is a constant equal to − B k + l k + l ) if k = 0 or l = 1 and 0 else, F l is the Faulhaberpolynomial of positive integer degree l , defined by F l ( n ) = (cid:80) ni =1 i l − for all n ∈ Z > , and the multiplicity r m ( λ ) of parts of size m in a partition λ is defined as the number of parts of λ of size m . The generating series T ( z ) = − u − v + (cid:80) k + l ≡ T k,l u k v (cid:96) − ( k )!( (cid:96) − satisfies [Itt20b,Theorem 4.4.1.] G ( u, v ) := (cid:104) T ( u, v ) (cid:105) q = −
12 Θ( u + v )Θ( u )Θ( v ) . Hence, the 1-point function G is a Jacobi form of weight 1 and index Q (( u, v ) , ( u, v )) = uv .This example now deviates from the previous ones because (cid:104) T ( u , v ) · · · T ( u n , v n ) (cid:105) q are notJacobi forms of some fixed weight (but rather a linear combination of functions of differentweights). It is, therefore, that we have to consider a different product (cid:12) on the space offunctions of partitions, for which G n ( u, v ) := (cid:104) T ( u , v ) (cid:12) · · · (cid:12) T ( u n , v n ) (cid:105) q = (cid:18) −
12 Θ( u + v )Θ( u )Θ( v ) (cid:19) n . (28)This product is uniquely determined by the isomorphism C P → C [ u , u , . . . ], denoted by f (cid:55)→ (cid:104) f (cid:105) u , by (cid:104) f (cid:12) g (cid:105) u = (cid:104) f (cid:105) u (cid:104) g (cid:105) u , where (cid:104) f (cid:105) u := (cid:80) λ ∈ P f ( λ ) u λ (cid:80) λ ∈ P u λ ( u λ = u λ u λ · · · ) . (29)The algebra T generated by the T k,l as defined before contains exactly the same elementsas the algebra consisting of polynomials in the functions T k,l with multiplication given by (cid:12) [Itt20b].Observing that in Section 3.1 one can replace the pointwise product on functions of parti-tions by (cid:12) , we have the following generalisations of the algebra T . (i). For a, b ∈ Q , and k ≥ , (cid:96) ≥
1, let T k,(cid:96) ( a, b ) = C k,(cid:96) ( a, b ) + ∞ (cid:88) m =1 m k (cid:0) e ( am ) F b(cid:96) ( r m ( λ )) + ( − k + l e ( − am ) F − b(cid:96) ( r m ( λ )) (cid:1) , where C k,(cid:96) ( a, b ) = α k ( a ) (cid:96) = 1 α (cid:96) − ( b ) k = 00 else,36ith the constants α k are defined in the previous section. Also, F b(cid:96) is the polynomial ofdegree (cid:96) − (cid:96) if b ∈ Z ) given by F b(cid:96) ( n ) = (cid:80) ni =1 e ( bi ) i l − for all n ∈ Z > . Letthe Q -algebra T ( N ) be generated by the functions T k,(cid:96) ( a, b ), where a, b ∈ N Z , under theinduced product . Assign to T k,(cid:96) ( a, b ) weight k + (cid:96) and extend to a weight grading under theinduced product. Let M = (2 , N ) N . Corollary 3.14.
The algebra T ( N ) is quasimodular of level M , after scaling τ by M . More concretely, for all homogeneous f ∈ T ( N ) of some weight k , the rescaled q -bracket (cid:104) f (cid:105) q M ,where q M = q /M , is a quasimodular form of weight k for Γ( M ). (ii). Let k , (cid:96) ∈ Z n and a , b ∈ N Z n . Corollary 3.15.
Whenever a · b ∈ Z + N Z , one has that (cid:104) T k ,(cid:96) ( a , b ) (cid:12) · · · (cid:12) T k n ,(cid:96) n ( a n , b n ) (cid:105) q is a quasimodular form of weight | k | + | (cid:96) | for Γ ( N ) . (iii). For k, (cid:96), s, t ∈ Z > let T ( s,t ) k,(cid:96) ( λ ) := C k,(cid:96) + (cid:88) m m k F (cid:96) (cid:16)(cid:106) r ms ( λ ) t (cid:107)(cid:17) . Denote by T ( N ) the algebra generated by the T ( s,t ) k,(cid:96) , where k, (cid:96) are even and s, t | N , underthe induced product . Assign to T ( s,t ) k,(cid:96) weight k + (cid:96) and extend to a weight grading under theinduced product. Corollary 3.16.
The algebra T ( N ) is quasimodular for Γ ( N ) . More concretely, for all homogeneous f ∈ H ( N ) of weight k , the q -bracket (cid:104) f (cid:105) q is a quasi-modular form of weight k for Γ ( N ). Remark . Combining this result with Corollary 3.15, by restricting to t = 1, we find thatany polynomial in T ( s, k,(cid:96) with respect to the induced product is quasimodular for Γ ( N ). Asimilar statement holds after restricting to s = 1. (cid:52)
4. When is the q -bracket modular? We first state and prove our answer to the question in the title of this section in full gener-ality, using the main result on the Taylor coefficients of quasi-Jacobi forms (Theorem 2.42).Afterwards we provide many examples, i.e. we prove the results on the ‘modular subspace’ ofthe Bloch–Okounkov algebra as stated in the introduction, and state similar results for theBloch–Okounkov algebra for congruence subgroups as well as the algebra of double momentfunctions. 37 .1. Construction of functions with modular q -bracket Given a quasi-Jacobi form ϕ of weight k satisfying the conditions of Theorem 2.42, thefunctions ξ (cid:96) ( ϕ ) = (cid:88) r ( − r (cid:88) s ≤ r ( D τ + e ) r Q r − s g (cid:96) ,s ( ϕ )( k + | (cid:96) | − r − ) r ( r − s )! , where e is given by Equation 15, and Q r − s g (cid:96) ,s ( ϕ ) in Definition 2.39, are modular forms (seealso Remark 2.40). Therefore, as in the previous section, assume that Φ : P × C r → C is suchthat for all n ≥ k ∈ Z such that the function ϕ Φ n : h × M n,r ( C ) → C given by ϕ Φ n ( τ, Z ) := (cid:68) n (cid:89) i =1 Φ( · , Z i ) (cid:69) q , where Z i is the i th row of Z , is a meromorphic quasi-Jacobi form of weight kn which admitsa Laurent expansion around all Z ∈ M n,r ( Q ) (after identifying M n,r ( C ) with C nr ). Write F = F Φ (1) for the graded algebra of Taylor coefficients of Φ (see Definition 3.2). Given (cid:96) ∈ Z n , our aim is to find h (cid:96) ∈ F such that (cid:104) h (cid:96) (cid:105) q = ξ (cid:96) ( ϕ n ), i.e., we want to determine a pullback of ξ (cid:96) ( ϕ n ) under the q -bracket.In order to do so, assume the algebra F satisfies the following two properties:(i) Q ∈ F ,(ii) There is a linear operator D acting on F such that (cid:68) D n (cid:89) i =1 Φ( z i ) (cid:69) q = ( d τ + z d z + . . . + z n d z n ) ϕ, (30)where D is extended to a linear operator on F [[ z , . . . , z n ]] by D ( f z (cid:96) ) = D ( f ) z (cid:96) forall f ∈ F and (cid:96) ∈ Z n . Remark . Recall D τ = π i ∂∂τ and e = π e = − (cid:80) m,r ≥ m q mr is the quasimodularEisenstein series of weight 2. The function Q is known to make the q -bracket equivariantwith respect to the operator D τ + e , i.e. (cid:104) Q f (cid:105) q = ( D + e ) (cid:104) f (cid:105) q for all f : P → C , explaining the first condition. The second condition is explained by notingthat g (cid:96) ,s ( ϕ ) = g (cid:96) (cid:16)(cid:88) i + | j | = s ϕ i, j z j (cid:17) = 1 s ! g (cid:96) (cid:0) ( d τ + z d z + . . . + z n d z n ) s ϕ (cid:1) . (31) (cid:52) Recall that the algebra F = F Φ (1) is generated by the Taylor coefficients f (cid:96) = f (cid:96) ( ) of Φfor (cid:96) ∈ Z r . An arbitrary monomial f L in F is given by f L · · · f L n with L ∈ M n,r ( Z ). Wedefine an operator on F corresponding to the index of ϕ . Definition 4.2.
Let M = ( m i,j ) ∈ M n ( Q ) be the index of ϕ Φ n . Let Q be the linear operatoron F which is given on monomials by Q f L = (cid:88) i,j m i,j f L − e i − e j ( L ∈ M n,r ( Z ) (cid:39) Z nr ) , where, on the right hand side, e i is a unit vector in Z nr .38ow, Theorem 1.6 can more explicitly be stated as follows, where the three propertiesbelow should be compared with the three properties satisfied by the functions h k in theintroduction (Section 1.2). Theorem 4.3.
Let F be a quasimodular algebra satisfying the above conditions, and Q , D : F → F the operators defined by Definition 4.2 and Equation 30, respectively. Then, thelinear mapping π : F → F given by π ( f ) = (cid:88) r ≥ r (cid:88) s =0 ( − r Q r Q r − s D s f ( m − r − ) r ( r − s )! s ! whenever f ∈ F is homogeneous of weight m satisfies (cid:104) π ( F ) (cid:105) q ⊆ M. Furthermore, one can choose a vector subspace
M ⊆ π F such that (i) F = M ⊕ Q F ; (ii) (cid:104)M(cid:105) q ⊆ M ; (iii) (cid:104) Q F (cid:105) q ∩ M = { } , where M denotes the algebra of modular forms for SL ( Z ) with rational Fourier coefficients .Remark . For the Bloch–Okounkov algebra Λ ∗ the mapping π turns out be the canoncialprojection of Λ ∗ on H in [Itt20a] and M = H . We do not expect that the conditions in thissection ensure that π is a projection in general, nor that π ( Q F ) = { } (in which case onecould choose M = π ( F )), nor that the splitting in (i) is canonical. However, just as with[Itt20a], once one has chosen M it follows immediately from (i) that every element of F hasa canonical expansion f = (cid:88) i ≥ f i Q i with f i ∈ M , and that (cid:104) f (cid:105) q ∈ M precisely if (cid:104) f i (cid:105) q = 0 for all i > (cid:52) Proof.
By (31) and by construction of Q one has that1 s ! (cid:10) Q r − s D s f m (cid:11) q = ( Q r − s g m )( ϕ ) s , from which it follows that (cid:104) h ( f m ) (cid:105) q = ξ m ( ϕ ) , where ϕ is the meromorphic Jacobi form ϕ ( z ) = (cid:104) (cid:81) ni =1 Φ( z i ) (cid:105) q . Hence, π ( f ) is modularunder the q -bracket for all f ∈ F .Choose M ⊆ F such that F = M ⊕ Q F . Then, we take M = π M . As, by definition, π ( f ) − f is a multiple of Q , the first property follows. The second property is immediate as M ⊆ π ( F ). For the last property, let f ∈ Q F with (cid:104) f (cid:105) q ∈ M be given. As f is divisibleby Q , the q -bracket (cid:104) f (cid:105) q is in the image of D + e acting on quasimodular forms. Now, thezero function is the only function in the image of D + e which is modular, so that the lastproperty follows.The rest of this chapter is devoted to examples of quasimodular algebras to which Theo-rem 4.3 applies. 39 .2. First example: the Bloch–Okounkov algebra In order to apply the results of the previous subsection to the Bloch–Okounkov algebra Λ ∗ , wehave to understand how the operators d τ and d z i act on the n -points functions F n (defined byDefinition 1.7), or equivalently, we have to understand the transformation behaviour of F n .This behaviour is uniquely determined by the following two properties. Proposition 4.5.
For all n ≥ one has d τ F n ( z , . . . , z n ) = 0 , d z F n ( z , . . . , z n ) = n (cid:88) i =2 F n − ( z + z i , z , . . . , ˆ z i , . . . , z n ) . Proof.
The second equality is equivalent to [BO00, Theorem 0.6], whereas the first statementseems not to be in the literature. As both statements follow by more or less the sameargument, we give both proofs. That is, we prove d τ V n ( z , . . . , z n ) = 0 , d z i V n ( z , . . . , z n ) = (cid:40) V n − ( z , . . . , z i + z i +1 , . . . , z n ) i < n i = n (32)inductively using the recursion (6), from which the proposition follows directly.For n = 1 both statements are clearly true. Hence, by the identity[ d τ , D mz ] = − mD m − z d z + m ( m − D m − z . and after assuming that d τ V n = 0, we find that θ ( z + . . . + z n +1 ) d τ V n +1 ( z ) equals − n − (cid:88) m =0 ( − n − − m ( n − − m )! θ ( n − − m ) ( z + . . . + z m ) V m ( z , . . . , z m ) . By the recursion (6) one finds that this expression vanishes. Hence, d τ V n +1 = 0 and d τ F n = 0as desired.Denote V im − ( z , . . . , z m ) = V m − ( z , . . . , z i + z i +1 , . . . , z m ) . By applying d z i to the recursion (6), using the identity [ d z , D mz ] = − mD m − z I (with I theindex operator, see Remark 2.28) and assuming that (32) holds, we find θ ( z + . . . + z n +1 ) d z i V n +1 ( z , . . . , z n +1 )= n (cid:88) m = i ( − n − m ( n − m )! θ ( n − m ) ( z + . . . + z m ) V m ( z , . . . , z m ) + − n (cid:88) m = i +1 ( − n +1 − m ( n + 1 − m )! θ ( n +1 − m ) ( z + . . . + z m ) V im − ( z , . . . , z m )= − i − (cid:88) m =0 ( − n − m ( n − m )! θ ( n − m ) ( z + . . . + z m ) V m ( z , . . . , z m ) +40 n (cid:88) m = i +1 ( − n +1 − m ( n + 1 − m )! θ ( n +1 − m ) ( z + . . . + z m ) V im − ( z , . . . , z m )= θ ( z + . . . + z n +1 ) V n ( z , . . . , z i + z i +1 , . . . , z n +1 ) . Next, recall the j th order differential operators D j in [Itt20a]. Definition 4.6.
Define the j th order differential operators D j by D j = (cid:88) i ∈ Z j ≥ (cid:18) | i | i , i , . . . , i j (cid:19) Q | i | ∂ i , with ∂ i = ∂ j ∂Q i +1 ∂Q i j +1 · · · ∂Q i j +1 , where the coefficient is a multinomial coefficient (in this section we pretend these operatorsact on Λ ∗ , although formally there are only defined on the formal algebra R = Q [ Q , Q , . . . ]freely generated by the variables Q , Q , . . . , which admits a canonical mapping to Λ ∗ .These operators turn out to correspond to certain symmetric powers of the derivativeoperators d z i . In particular, observe that the coefficient of z (cid:96) in the case j = 1 below,is given by g (cid:96) ( F n ) . Hence, the operator D , requested in the previous section, is given by D = D / Proposition 4.7.
For all j ≥ one has (cid:104) D j W ( z ) (cid:105) q = j ! (cid:0) z d j − z + . . . + z n d j − z n (cid:1) F n ( z ) . Proof.
Observe that D j Q (cid:96) = j ! (cid:88) i <...
Equation 33, as well as the fact that π is a projection, follows directly from [Itt20a,Corollary 2]. In particular, as ∆ ( Q / ) = 0, it follows that π ( Q f ) = 0.For the properties of M , observe that D and Q , defined in the previous section, can also beexpressed as D / − ∂ / { π ( Q λ ) } , where λ goes over all partition with all parts atleast 3, is a basis for both spaces. In order to extend the result in the previous subsection to the Bloch–Okounkov algebras Λ ∗ ( N )of level N (see (4) and Section 3.2), we should generalise the operators ∂ and D . Definition 4.9.
Let (cid:98) R = Q [ Q k ( a ) | k ∈ Z , a ∈ Q ] be the algebra in the formal variables Q k ( a )with canonical projection to (cid:83) N ∈ Z Λ ∗ ( N ). Given a ∈ Q j , define the j th order differentialoperators D j on ˆ R by D j = (cid:88) i ∈ Z j ≥ (cid:88) a ∈ Q j (cid:18) | i | i , i , . . . , i j (cid:19) Q | i | ( | a | ) ∂ i ( a ) , where ∂ i ( a ) = ∂ j ∂Q i +1 ( a ) · · · ∂Q i n +1 ( a n ) . From now on we pretend that these operators act on Λ ∗ ( N ), by identifying Λ ∗ ( N ) witha quotient of ˆ R via the obvious inclusion map. Note that restricted to Λ ∗ the operators D j are the same as defined in Definition 4.6. Similarly, the operators D j satisfy the followingproperty. 42 roposition 4.10. For all j ≥ and a ∈ Q n one has (cid:104) D j W ( z + a ) (cid:105) q = j ! (cid:0) ( z + a ) d j − z + . . . + ( z n + a n ) d j − z n (cid:1) F n ( z + a ) . Therefore, Theorem 4.3 now specializes to the following result.
Theorem 4.11.
Let N ≥ and ∂ = D and D : Λ ∗ ( N ) → Λ ∗ ( N ) be given by Equation 4.9.Let the projection π : Λ ∗ ( N ) → Λ ∗ ( N ) be given by π ( f ) = (cid:88) r ≥ r (cid:88) s =0 ( − s Q r ∂ r − s D s ( f )2 r ( (cid:96) − r − ) r ( r − s )! s ! , whenever f ∈ Λ ∗ ( N ) is homogeneous of weight (cid:96) . Then, the subspace M ( N ) := π (Λ ∗ ( N )) of Λ ∗ ( N ) satisfies (i) Λ ∗ ( N ) = M ( N ) ⊕ Q Λ ∗ ( N ) ; (ii) (cid:104)M ( N ) (cid:105) q ⊆ M (Γ( N )) ; (iii) (cid:104) Q Λ ∗ ( N ) (cid:105) q ∩ M (Γ( N )) = { } , where M ( N ) denotes the algebra of modular forms of level N . For the algebra of double moments functions (see Section 3.5) the n -point functions withrespect to the induced product (cid:12) (see (29)) are given by (28), i.e., G n ( u, v ) = (cid:16) Θ( u + v )Θ( u )Θ( v ) (cid:17) n , which is a Jacobi form. Hence, the operators d τ , d u and d v vanish acting on G n , so that D can taken to be the zero operator. We write d for the derivation on T given by d ( T k,(cid:96) ) = k ( (cid:96) − T k − ,(cid:96) − − δ k + (cid:96) − , d ( f (cid:12) g ) = d ( f ) (cid:12) g + f (cid:12) d ( g )for all f, g ∈ T . This notation is suggested by the fact that (cid:104) d f (cid:105) q = d τ (cid:104) f (cid:105) q for all f ∈ T . Theorem 4.12.
Let the projection π : T → T be given by π ( f ) = (cid:88) r ≥ r r ! r (cid:122) (cid:125)(cid:124) (cid:123) T , (cid:12) · · · (cid:12) T , (cid:12) d r ( f ) . Then M = π ( T ) satisfies the following three properties: (i) T = M ⊕ ( T , ) , where ( T , ) = T , (cid:12) T ; (ii) (cid:104)M(cid:105) q = M ; (iii) (cid:104) T , (cid:12) T (cid:105) q ∩ M = { } . roof. The statement follows along the same lines as Theorem 4.3, by making the followingobservations: • Analogous to ξ (cid:96) ( ϕ ), the functions (cid:88) r ≤| (cid:96) | ( − r (cid:88) s ≤ r e r ( Q r − s g (cid:96) )( ϕ ) s ( r − s )!are modular forms exactly if ϕ is a quasi-Jacobi form; • (cid:104) T , (cid:12) f (cid:105) q = − e (cid:104) f (cid:105) q for all f ∈ C P ; • The operator d coincides with the operator Q . • By [Itt20b, Theorem 3.4.1] we have that (cid:104)T (cid:105) q = (cid:102) M , from which it follows that equalityholds in (ii). Remark . In fact, for all f, g ∈ T one has π ( f (cid:12) g ) = π ( f ) (cid:12) π ( g ) . Hence, (cid:104) π T (cid:105) q is uniquely determined by (cid:104) π ( T , ) (cid:105) q = (cid:104) π ( T , ) (cid:105) q = 0 and (cid:104) π ( T k,l ) (cid:105) q = (cid:40) ϑ k − G l − k +2 l ≥ kϑ l G k − l k ≥ l + 2for T k,l ∈ T with k + l ≥
4, where ϑ denotes the Serre derivative. The derivatives of Eisensteinseries appearing on the right of this equation, with the case distinction according to the signof l + 1 − k , should be compared to the Taylor coefficients of G ( u, v ) in [Zag91]. Moreover,they are very similar (but here in level 1 and there in level 2, and here with Serre derivativesand there with usual derivatives) to the ones that appeared in [KZ95] to prove the originalassertion of Dijkgraaf from which the whole Bloch–Okounkov story arose. (cid:52) A. The action of the Jacobi group
We prove Proposition 2.4, or, more precisely, we show that the real Jacobi group acts on allmeromorphic functions ϕ : h × C n → C .In order to define the real Jacobi group, we let ω : M ,n ( R ) × M n, ( R ) → Sym n ( R ) ι : M ,n ( R ) × M ,n ( R ) → Skew n ( R )( X, Y ) (cid:55)→ − X t J Y + Y t J X ( X, Y ) (cid:55)→ X t J Y, where J = (cid:0) −
11 0 (cid:1) and Sym n and Skew n denote the space of symmetric and skew-symmetric n × n -matrices respectively. Then, we define H n to be the central extension of M ,n ( R ) by Sym n ( R )corresponding to the 2-cocycle ω , so that H n = { ( X, κ ) : X ∈ M ,n ( R ) , κ + ι ( X, X ) ∈ Sym n ( R ) } with multiplication ( X, κ )( X (cid:48) , κ (cid:48) ) = ( X + X (cid:48) , κ + κ (cid:48) − ι ( X, X (cid:48) )) . emark A.1 . Observe that if n = 1, this group H is isomorphic to the Heisenberg group H R with underlying set R × R and product given by( X, κ )( X (cid:48) , κ (cid:48) ) = ( X + X (cid:48) , κ + κ (cid:48) + det( X, X (cid:48) )) . (cid:52) Definition A.2.
The real Jacobi group is defined as G Jn = SL ( R ) (cid:110) H n with respect to theaction of SL ( R ) on H n given by γ ( X, κ ) = ( γX, κ ).That is, G Jn = (cid:8) ( γ, X, κ ) : γ ∈ SL ( R ) , X ∈ M ,n ( R ) , κ + ι ( X, X ) ∈ Sym n ( R ) (cid:9) with multiplication given by( γ, X, κ )( γ (cid:48) , X (cid:48) , κ (cid:48) ) = ( γγ (cid:48) , γ (cid:48) X + X (cid:48) , κ + κ (cid:48) − ι ( γ (cid:48) X, X (cid:48) )) . Under the natural inclusion, we can think of γ ∈ SL ( R ) , X ∈ M ,n ( R ) and κ ∈ M n ( R )as elements of the real Jacobi group G Jn . In order to avoid confusion with Definition 2.2,we write (cid:111) for this slash operation of G Jn . After the definition we explain why this action iswell-defined. Definition A.3.
Given an integer k and M ∈ M n ( Q ), the real Jacobi group G Jn acts onthe space of meromorphic functions ϕ : h × C n → C , where the action for ( γ, X, κ ) ∈ G Jn isuniquely determined by(i) ( ϕ (cid:111) k,M γ )( τ, z ) := ( cτ + d ) − k e (cid:16) − c Q M ( z ) cτ + d (cid:17) ϕ (cid:16) aτ + bcτ + d , z cτ + d (cid:17) (cid:0) γ = (cid:0) a bc d (cid:1)(cid:1) (ii) ( ϕ (cid:111) M X )( τ, z ) := e ( B M ( λ , λ τ + 2 z + µ )) ϕ ( τ, z + λ τ + µ ) (cid:0) X = (cid:0) λµ (cid:1)(cid:1) (iii) ( ϕ (cid:111) M κ )( τ, z ) := e ( (cid:104) M, κ (cid:105) ) ϕ ( τ, z )where (cid:104) M, κ (cid:105) denotes the Frobenius inner product of M = ( m i,j ) and κ , i.e., (cid:104) M, κ (cid:105) = (cid:88) i,j m i,j κ i,j . Remark
A.4 . The Frobenius inner product can easily be expressed in terms of the bilinearform, in the following way. For all M ∈ M n ( Q ) and X, X (cid:48) ∈ M ,n ( Q ) one has by a directcomputation that (cid:104) M, ι ( X, X (cid:48) ) (cid:105) = B M ( λ (cid:48) , µ ) − B M ( λ , µ (cid:48) ) , (34)where X = (cid:0) λµ (cid:1) and X (cid:48) = (cid:0) λ (cid:48) µ (cid:48) (cid:1) . (cid:52) We now show that the above action of the real Jacobi group G Jn on the space of meromorphicfunctions ϕ : h × C n → C by the above slash equation. It is clear that the identity (1 , ,
0) actstrivially. Hence, it suffices to prove the following identities for all ( γ, X, κ ) , ( γ (cid:48) , X (cid:48) , κ (cid:48) ) ∈ G Jn ϕ (cid:111) γ (cid:111) γ (cid:48) = ϕ (cid:111) ( γγ (cid:48) );2. ϕ (cid:111) X (cid:111) X (cid:48) = ϕ (cid:111) ( X + X (cid:48) ) (cid:111) − ι ( X, X (cid:48) ); 45. ϕ (cid:111) κ (cid:111) κ (cid:48) = ϕ (cid:111) ( κ + κ (cid:48) );4. ϕ (cid:111) X (cid:111) γ = ϕ (cid:111) γ (cid:111) ( γX );5. ϕ (cid:111) κ (cid:111) γ = ϕ (cid:111) γ (cid:111) κ ;6. ϕ (cid:111) κ (cid:111) X = ϕ (cid:111) X (cid:111) κ .The first identity follows by observing that the automorphic factor j ( γ, z ) = cτ + d and thequasimodular factor w ( γ, z ) = ccτ + d (where γ = (cid:0) a bc d (cid:1) ) satisfy j ( γγ (cid:48) , τ ) = j ( γ (cid:48) , τ ) j ( γ, γ (cid:48) τ ) , w ( γγ (cid:48) , τ ) = j ( γ (cid:48) , τ ) − w ( γ, γ (cid:48) τ ) + w ( γ (cid:48) , τ ) . The second identity follows from (34). The third follows by linearity of the inner product.Writing X = (cid:0) λµ (cid:1) and γX = ( λ (cid:48) , µ (cid:48) ), we find that the fourth equation is a consequence ofthe identity z (cid:111) X (cid:111) , γ = z (cid:111) γ (cid:111) ( γX )and the fact that − w ( γ, τ ) Q M ( z ) + B M ( λ , λ ( γτ ) + 2 z j ( γ, τ ) − + µ )equals − w ( γ, τ ) Q M ( z + λ (cid:48) τ + µ (cid:48) ) + B M ( λ (cid:48) , λ (cid:48) τ + 2 z + µ (cid:48) )for all M ∈ M n ( Q ).The last two equations follow directly as the slash action with κ multiplies ϕ with a constantnot depending on z and τ .Let C Z = { (1 , , κ ) | κ ∈ Sym n ( Z ) } . We fix the group homomorphism ι : Γ Jn → G Jn /C Z given by( γ, X ) (cid:55)→ ( γ, X, . Observe that C Z acts trivially by the above action in case Q M is an integer quadratic form.Hence, in this case we have an action of Γ Jn on functions h × C → C . In case Q M is not aninteger quadratic form we modify the slash action by a character of G Jn : Proposition A.5.
Given M ∈ M n ( Q ) , there exist a character ρ : G Jn → C × satisfying ρ ( γ ) =1 for γ ∈ SL ( Z ) and such that ( ϕ, g ) (cid:55)→ ρ ( g ) ϕ (cid:111) k,M g defines an action of Γ Jn on the space of functions ϕ : h × C n → C . Moreover, when κ ∈ Sym n ( Z ) the character ρ is given by ρ ( γ, X, κ ) = e ( Q M ( λ ) − B M ( λ , µ ) + Q M ( µ ) + (cid:104) M, κ (cid:105) ) , (35) where X = (cid:0) λµ (cid:1) . roof. To prove that ρ satisfies (35), we first show that ρ is uniquely determined whenever κ ∈ Sym n ( Z ). Note that ρ ( κ ) = e ( (cid:104) M, κ (cid:105) ) for all κ ∈ Sym n ( Z ) as C Z should act trivially. Becauseof the assumption ρ ( γ ) = 1, it suffices to show that ρ is uniquely determined on M ,n ( R ).Now, note that for λ ∈ R n one has ρ ( λ , ) = ρ ( λ , ) ρ (( )) = ρ ( λ , λ ) = ρ ( λ , ) ρ ( , λ ) ρ ( − ι (( λ , ) , ( , λ ))) . Hence, ρ ( λ , ) = ρ ( λ , ) ρ ( (cid:0) − (cid:1) ) = ρ ( , λ ) = ρ ( ι (( λ , ) , ( , λ ))) . This shows ρ is uniquely determined on the generators of M ,n ( R ).Next, we show that ρ as defined by (35) is a group homomorphism Γ Jn → C × . Observethat ρ ( κ ) ϕ (cid:111) κ = e (2 (cid:104) M, κ (cid:105) ) ϕ, so ρ is constant on C Z . Hence, that ρ is a group homomorphism follows directly if weshow that ρ ( X ) = ρ ( γX ) and ρ ( X ) ρ ( X (cid:48) ) = ρ ( X + X (cid:48) ) ρ ( − ι ( X, X (cid:48) )) for all γ ∈ SL ( Z )and X, X (cid:48) ∈ M ,n ( Z ).For the first write γ = (cid:0) a bc d (cid:1) . Then ρ ( γX ) = e (( a + b − ab ) Q M ( λ ) + (2 ac + 2 bd − ad − bc ) B M ( λ , µ ) + ( c + d − cd ) Q M ( µ )) . As not both a and b are even, it follows that a + b − ab ≡ c + d − cd ≡ ac + 2 bd − ac − bc ≡ ad − bc = 1. Now, as the bilinear form Q M takes values in Z , it follows that ρ ( γX ) = ρ ( X ).The second equation follows from (34), namely, using Q M takes half-integral values oninteger vector, one has ρ ( X ) ρ ( X (cid:48) ) = e ( Q M ( λ ) − B M ( λ , µ ) + Q M ( µ ) + Q M ( λ (cid:48) ) − B M ( λ (cid:48) , µ (cid:48) ) + Q M ( µ (cid:48) ))= e ( B M ( λ + λ (cid:48) , λ + λ (cid:48) ) + B M ( µ + µ (cid:48) , µ + µ (cid:48) ) − B M ( λ + λ (cid:48) , µ + µ (cid:48) ) ++ B M ( λ , µ (cid:48) ) + B M ( λ (cid:48) , µ ) )= ρ ( X + X (cid:48) ) ρ ( − ι ( X, X (cid:48) )) . Definition A.6.
Given M ∈ M n ( Q ), we define the action of Γ Jn on a function ϕ : h × C n → C by the above action ϕ | k,M g = ρ ( g ) ϕ (cid:111) k,M g under the group homomorphism ι : Γ Jn → G Jn /C Z . Note that this definition coincides withDefinition 2.2. Proof of Proposition 2.4.
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