On mod $p$ singular modular forms

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 $\mathfrak{p}$ of $K$, is a constant modulo $\mathfrak{p}$. A similar property also holds for Siegel modular forms. Moreover, we define the notion of mod $\mathfrak{p}$ singular modular forms and discuss some relations between their weights and the corresponding prime $p$. We discuss some examples of mod $\mathfrak{p}$ singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms. Finally, we apply our results to properties mod $\mathfrak{p}$ of Klingen-Eisenstein series.

21 pages

Cited by in corpus (2)

• Note on mod p property of Hermitian modular forms
• Notes on theta series for Niemeier lattices II