Paramodular cusp forms

Poor, Cris; Yuen, David

doi:10.1090/S0025-5718-2014-02870-6

Paramodular cusp forms
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by Cris Poor and David S. Yuen PDF

Math. Comp. 84 (2015), 1401-1438 Request permission

Abstract:

We classify Siegel modular cusp forms of weight two for the paramodular group $K(p)$ for primes $p< 600$. We find evidence that rational weight two Hecke eigenforms beyond the Gritsenko lifts correspond to certain abelian surfaces defined over $\mathbb {Q}$ of conductor $p$. The arithmetic classification is in the companion article by A. Brumer and K. Kramer, Paramodular abelian varieties of odd conductor. The Paramodular Conjecture, supported by these computations and consistent with the Langlands philosophy and the work of H. Yoshida, Siegel’s modular forms and the arithmetic of quadratic forms, is a partial extension to degree two 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.

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