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Mathematics of Computation

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Paramodular cusp forms

Authors: Cris Poor and David S. Yuen
Journal: Math. Comp. 84 (2015), 1401-1438
MSC (2010): Primary 11F46; Secondary 11F50
Published electronically: August 20, 2014
MathSciNet review: 3315514
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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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Cris Poor
Affiliation: Department of Mathematics, Fordham University, Bronx, New York 10458

David S. Yuen
Affiliation: Mathematics and Computer Science Department, Lake Forest College, 555 N. Sheridan Road, Lake Forest, Illinois 60045

Keywords: Paramodular, Hecke eigenform
Received by editor(s): April 19, 2012
Received by editor(s) in revised form: June 14, 2013, and August 6, 2013
Published electronically: August 20, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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