On mod $p$ singular modular forms

TL;DR

This paper proves that elliptic and Siegel modular forms with Fourier coefficients divisible by a prime ideal are constant mod that prime, introduces mod p singular modular forms, and explores their properties and examples.

Contribution

It establishes the constancy of modular forms with almost all Fourier coefficients divisible by a prime ideal and introduces the concept of mod p singular modular forms with related properties.

Findings

Modular forms with almost all Fourier coefficients divisible by a prime are constant mod that prime.

Introduces the notion of mod p singular modular forms and explores their weight relations.

Provides examples from Eisenstein and theta series, and studies properties of Klingen-Eisenstein series mod p.

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.