On the kernel of the theta operator mod p

Abstract

We construct many examples of level one Siegel modular forms in the kernel of theta operators mod p by using theta series attached to positive definite quadratic forms.

Weights of the Mod p Kernels of Theta Operators

“…In [6], we use certain theta series attached to quadratic forms to construct several types of modular forms in the kernel of theta operators mod p. We compute ω N for some cases, here N can be an arbitrary number coprime to p. In this way we confirm that the constructions in [6] are the best possible ones in the sense that the level one forms obtained are of smallest possible weight.…”

“…all automorphisms have determinant +1). Then it was shown in [6] that this theta series is congruent to a (cuspidal) level one modular form F of weight…”

“…After this, Mizumoto [21] found another example of weight 16 and p = 31, which comes from the Klingen-Eisenstein series arising from a cusp form of weight 16. Recently, the authors, Kodama and Nagaoka [6,17,22,25] ) and degree n if the weight is even (resp. odd).…”

Theta operator on Hermitian modular forms over the Eisenstein field

“…The first attempt to generalize Ramanujan's theta operator to a higher degree case was made in [1]. Subsequently, this generalization was developed by several researchers (e.g., [7], [2], [8]). In the case of Siegel modular forms, the theta operator Θ is defined by F = a F (T )q T −→ Θ(F ) := det(T ) · a F (T )q T for the generalized q-expansion F = a F (T )q T .…”

“…For example, Igusa's cusp form χ 35 is an element of the mod 23 kernel of the theta operator as stated above. Moreover, the Siegel Eisenstein series E (2) 12 and the Siegel theta series ϑ (2) L associated with the Leech lattice L satisfy…”