Siegel modular forms mod p

TL;DR

This paper proves a vanishing theorem for one forms on moduli stacks of abelian varieties in characteristic p, enabling computation of Picard groups and revealing new congruences between Siegel modular forms.

Contribution

It introduces a vanishing theorem for one forms in characteristic p, facilitating Picard group calculations and uncovering novel congruences between modular forms.

Findings

Vanishing theorem for one forms on moduli stacks in characteristic p

Calculation of Picard groups of these stacks

New congruences between integral Siegel modular forms

Abstract

We prove a vanishing theorem for one forms on the moduli stack of principally polarized abelian varieties of genus g>1 with level structure N over fields of characteristic p different from two. This is used to compute the Picard groups of these stacks. As an application we obtain results one congruences between integral Siegel modular forms.