On mod p singular modular forms

Abstract

Abstract We show that a Siegel modular form with integral Fourier coefficients in a number field K, for which all but finitely many coefficients (up to equivalence) are divisible by a prime ideal 𝔭 ${\mathfrak{p}}$ of K, is a constant modulo 𝔭 ${\mathfrak{p}}$ . Moreover, we define a notion of mod 𝔭 ${\mathfrak{p}}$ singular modular form and discuss a relation between its weight and the corresponding prime p. We discuss some examples of mod 𝔭 ${\mathfrak{p}}$ singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms. Finally, we apply our results to properties mod 𝔭 ${\mathfrak{p}}$ of Klingen–Eisenstein series.