Paramodular Cusp Forms

TL;DR

This paper classifies certain Siegel modular cusp forms of weight two for the paramodular group, linking nonlift forms to abelian varieties and supporting a conjecture extending the Shimura-Taniyama framework.

Contribution

It provides a classification of weight two paramodular cusp forms and establishes a connection between nonlift forms and abelian varieties, supporting the Paramodular Conjecture.

Findings

Weight two nonlift Hecke eigenforms correspond to abelian varieties over rationals.

These forms share Euler factors with associated abelian varieties.

They satisfy congruences with Gritsenko lifts under certain torsion conditions.

Abstract

We classify Siegel modular cusp forms of weight two for the paramodular group K(p) for primes p< 600. We find that weight two Hecke eigenforms beyond the Gritsenko lifts correspond to certain abelian varieties defined over the rationals of conductor p. The arithmetic classification is in a companion article by A. Brumer and K. Kramer. The Paramodular Conjecture, supported by these computations and consistent with the Langlands philosophy and the work of H. Yoshida, is a partial extension to degree 2 of the Shimura-Taniyama Conjecture. These nonlift Hecke eigenforms share Euler factors with the corresponding abelian variety $A$ and satisfy congruences modulo \ell with Gritsenko lifts, whenever $A$ has rational \ell-torsion.