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arXiv · 1305.2813

On mod $p$ singular modular forms

Siegfried Böcherer·Toshiyuki Kikuta
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Abstract

We show that an elliptic modular form with integral Fourier coefficients in a number field $K$, for which all but finitely many coefficients are divisible by a prime ideal $\frak{p}$ of $K$, is a constant modulo $\frak{p}$. A similar property also holds for Siegel modular forms. Moreover, we define the notion of mod $\frak{p}$ singular modular forms and discuss some relations between their weights and the corresponding prime $p$. We discuss some examples of mod $\frak{p}$ singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms. Finally, we apply our results to properties mod $\frak{p}$ of Klingen-Eisenstein series.

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Siegfried Böcherer, Toshiyuki Kikuta. 2013-05-13. On mod $p$ singular modular forms. https://arxiv.org/abs/1305.2813

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