Weights of the Mod p Kernels of Theta Operators

Abstract

We give some relations between the weights and the prime p of elements of the mod p kernel of the generalized theta operator Θ [j] . In order to construct examples of the mod p kernel of Θ [j] from any modular form, we introduce new operators A (j) (M ) and show the modularity of F |A (j) (M ) when F is a modular form. Finally, we give some examples of the mod p kernel of Θ [j] and the filtrations of some of them.

On the mod p kernel of the theta operator and Eisenstein series

“…(3) There are several modular forms in M 12 (Γ (2) ) satisfying a congruence relation similar to that given in (2). For example,…”

“…(2) For the case that m = 2, the mod p vanishing property of Θ(E (2) k ) has previously been studied (Kikuta-Nagaoka [14]).…”

On the kernel of the theta operator mod p

“…We have to make an important comment on what we mean by "explicit construction" here: The kernel of Θ [j] mod p is a notion which depends only on modular forms mod p, therefore the weight of the constructed modular form is only of interest mod (p − 1). On the other hand, one is also interested in explicit small weights for which we can get modular forms in the kernel mod p. In this paper we address both versions of explicit construction, we will call them "weak construction" and "strong construction" respectively; in most cases our "strong construction" also gives the smallest possible weight, which is called "filtration" in the work of Serre and Swinnerton-Dyer, see [11] for details. In the final section we also show that some of the known examples of congruences for degree two Siegel modular forms can be explained by our methods.…”