On the restriction map for Jacobi forms

Abstract

In this article we give a description of the kernel of the restriction map for Jacobi forms of index 2 and obtain the injectivity of D 0 ⊕ D 2 on the space of Jacobi forms of weight 2 and index 2. We also obtain certain generalization of these results on certain subspace of Jacobi forms of square-free index m.

Taylor expansions of Jacobi forms and linear relations among theta series

“…In this direction, for the integral-index setting, there are some good works sharpening the original reslut of Eichler and Zagier. For instance, consult [Kra86], [AB99], [RS13], [DR15] and [DP17]. We wish that, their ideas could also be used to deal with lattice-index case.…”

Jacobi forms and differential operators: odd weights

“…When the index m = 1, the question about ker D 0 , which is nothing but the restriction map from J k,m (N) to M k (N), the space of elliptic modular forms of weight k on Γ 0 (N); defined by φ(τ, z) → φ(τ, 0), translates into the possibility of removing the differential operator D 2 while preserving injectivity. This question is also interesting in its own right, and has received some attention in the recent past, see the works [1,2,3,9], and the introduction there. We only note here that first results along this line of investigation seems to be by J. Kramer, who gave an explicit description of ker D 0 when m = 1 and level N a prime, in terms of the vanishing order of cusp forms in a certain subspace of S 4 (N) (This is related to the so-called Weierstrass subspaces of S k (N), see [2]).…”

“…We take this opportunity to note that the proof of [3, Theorem 1.2 (i)] is not correct as it is; however we stress that this does not affect any other result of the paper, moreover moreover this part of the result was already known before from [9].…”