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# Weights of the mod $p$ kernel of the theta operators

Published:2017-07-31
Printed: Apr 2018
• Siegfried Böcherer,
Mathematisches Institut, Universität Mannheim, 68131 Mannheim, Germany
• Toshiyuki Kikuta,
Faculty of Information Engineering, Department of Information and Systems Engineering, Fukuoka Institute of Technology, 3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka 811-0295, Japan
• Sho Takemori,
Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, 060-0810, Japan
 Format: LaTeX MathJax PDF

## Abstract

Let $\Theta ^{[j]}$ be an analogue of the Ramanujan theta operator for Siegel modular forms. For a given prime $p$, we give the weights of elements of mod $p$ kernel of $\Theta ^{[j]}$, where the mod $p$ kernel of $\Theta ^{[j]}$ is the set of all Siegel modular forms $F$ such that $\Theta ^{[j]}(F)$ is congruent to zero modulo $p$. In order to construct examples of the mod $p$ kernel of $\Theta ^{[j]}$ from any Siegel modular form, we introduce new operators $A^{(j)}(M)$ and show the modularity of $F|A^{(j)}(M)$ when $F$ is a Siegel modular form. Finally, we give some examples of the mod $p$ kernel of $\Theta ^{[j]}$ and the filtrations of some of them.
 Keywords: Siegel modular form, congruences for modular forms, Fourier coefficients, Ramanujan's operator, filtration
 MSC Classifications: 11F33 - Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 11F46 - Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

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