Weights of the Mod p Kernels of Theta Operators

Böcherer, Siegfried ; Kikuta, Toshiyuki ; Takemori, Sho

Canadian journal of mathematics, Tome 70 (2018) no. 2, pp. 241-264

Résumé

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.

Détail

DOI : 10.4153/CJM-2017-014-0

Mots-clés : 11F33, 11F46, Siegel modular form, congruences for modular forms, Fourier coefficients, Ramanujan's operator, filtration