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Abstract: We classify Siegel modular cusp forms of weight two for the paramodular groupKp for primes p< 600. We find that weight two Hecke eigenforms beyond theGritsenko lifts correspond to certain abelian varieties defined over therationals of conductor p. The arithmetic classification is in a companionarticle by A. Brumer and K. Kramer. The Paramodular Conjecture, supported bythese computations and consistent with the Langlands philosophy and the work ofH. Yoshida, is a partial extension to degree 2 of the Shimura-TaniyamaConjecture. These nonlift Hecke eigenforms share Euler factors with thecorresponding abelian variety $A$ and satisfy congruences modulo \ell withGritsenko lifts, whenever $A$ has rational \ell-torsion.

Author: Cris Poor, David S. Yuen


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