Congruences between Siegel modular forms

Abstract

Using the moduli theory of abelian varieties and a recent result of Böcherer-Nagaoka on lifting of the generalized Hasse invariant, we show congruences between the weights of Siegel modular forms with congruent Fourier expansions. This result implies that the weights of p-adic Siegel modular forms are well defined.

Weights of the Mod p Kernels of Theta Operators

“…It follows from [8,15] immediately that Proposition 2.1. Let p be an odd prime, N a positive integer with p ∤ N and…”

Congruences for Hermitian modular forms of degree 2

“…Ichikawa [7] showed that Serre's p-adic weight is well defined in the case of Siegel modular forms. We introduce the result in the general degree case.…”

“…Serre [15] defined the notion of p-adic modular forms and applied it to the construction of p-adic L functions. Some mathematicians have attempted to generalize the theory of Serre's p-adic modular forms to the case of several variables [3,7,11,13]. In particular, Ichikawa [7] showed that the p-adic weight is well-defined for Siegel modular forms of general degree.…”

On p-Adic Properties of Siegel Modular Forms

“…Starting with Swinnerton-Dyer [24] and Serre [18], the mod p properties of elliptic modular forms and also their p-adic properties have been deeply studied. Some aspects of this theory were later generalized to other types of modular forms like Jacobi forms [22] and also Siegel modular forms [14]. In our previous works we constructed Siegel modular form congruent 1 mod p; we did this for level one [6] and also for level p with additional good p-adic behavior in the other cusps [7].…”