Abstract view
Weights of the mod $p$ kernel of the theta operators


Siegfried BĂ¶cherer,
Mathematisches Institut, UniversitĂ¤t Mannheim, 68131 Mannheim, Germany
Toshiyuki Kikuta,
Faculty of Information Engineering, Department of Information and Systems Engineering, Fukuoka Institute of Technology, 3301 Wajirohigashi, Higashiku, Fukuoka 8110295, Japan
Sho Takemori,
Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, KitaKu, Sapporo, 0600810, Japan
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 $FA^{(j)}(M)$ 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.