The action of the heat operator on Jacobi forms

Abstract

Abstract. We investigate the action of the heat operator on Jacobi forms. In particular, we present two explicit characterizations of this action on Jacobi forms of index 1. Furthermore, we study congruences and filtrations of Jacobi forms. As an application, we determine when an analog of Atkin's U -operator applied to a Jacobi form is nonzero modulo a prime.

The structure of Siegel modular forms modulo $p$ and $U(p)$ congruences

“…Tate's theory of theta cycles (see §7 of [14]) relies on Lemma 5 of [31], which gives the filtration of the theta operator applied to a modular form. Proposition 2 of [26] extends Lemma 5 of [31] to Jacobi forms on H × C, and our next proposition extends this further to Jacobi forms on H × C l . Proof.…”

“…Proof. We proceed exactly as in [26], and we assume that ω(φ) = k. Let if ω L m (φ) < k + p + 1, then ω (2k−l) det(M ) 6 φE 2 < k + p + 1 by (2.9). It remains to show that ω (φE 2 ) = k + p + 1, which then implies that p divides (2k − l) det(2M ).…”

On congruences of Jacobi forms

“…In [12], we study the action of the heat operator L := on Jacobi forms of index 1. If φ is such a Jacobi form, then L n (φ) (for all n ∈ N) is a quasi-Jacobi form in the sense of Kawai and Yoshioka [9].…”

“…It is not important for our purposes, but Dφ can be explicitly characterized (for details, see [12]). For φ ∈ J k,1 , define the sequence φ r ∈ J k+2r,1 recursively by…”

“…Let φ 10,1 = 1 144 (E 6 E 4,1 − E 4 E 6,1 ) be the unique (up to a scalar) Jacobi cusp form of weight 10 and index 1. Recall Ramanujan's [11] identities and their Jacobi form generalizations in [12]: Hence Ω L 11 (φ 10,1 ) = 20 and φ 10,1 U 13 ≡ 0 (mod 13). Furthermore, Proposition 3 implies that φ 10,1 U p ≡ 0 (mod p) for all primes p > 13.…”