All Siegel Hecke eigensystems (mod $p$) are cuspidal

Fix integers $g\geq 1$ and $N\geq 3$, and a prime $p$ not dividing $N$. We show that the systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms (mod $p$) of dimension $g$, level $N$, and varying weight, are the same as the systems occurring in the spaces of Siegel \emph{cusp forms} with the same parameters and varying weight. In particular, in the case $g=1$, this says that the Hecke eigensystems (mod $p$) coming from classical modular forms are the same as those coming from cusp forms. The proof uses both the main theorem of~\cite{Ghitza2004a} and a modification of the techniques used there, namely restriction to the superspecial locus.

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Published 1 January 2006