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Weights of the Mod p Kernels of Theta Operators

Published online by Cambridge University Press:  20 November 2018

Siegfried Böcherer
Affiliation:
Mathematisches Institut, Universität Mannheim, 68131 Mannheim, Germany e-mail: boecherer@t-online.de
Toshiyuki Kikuta
Affiliation:
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 e-mail: kikuta@fit.ac.jp
Sho Takemori
Affiliation:
Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, 060-0810, Japan e-mail: takemori@math.sci.hokudai.ac.jp

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 forms, we introduce new operators ${{A}^{(j)}}(M)$ and show the modularity of $F|{{A}^{\left( j \right)}}\left( M \right)$ 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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

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