Paramodular Cusp Forms

2009/12/01 by Cris Poor, Poor, Cris, David S. Yuen +1 · 1 citation
Mathematics · #11F46 #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.0912.0049

openalex publication_date 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℓ with Gritsenko lifts, whenever A has rational ℓ-torsion.

Cited by

• Modularity theorems for abelian surfaces
• On the cohomology of congruence subgroups of SL₄ (\Z)

Related

• Nonlift weight two paramodular eigenform constructions
• Antisymmetric Paramodular Forms of Weights 2 and 3
• Quinary forms and paramodular forms
• Siegel modular forms of degree two attached to Hilbert modular forms
• Residual paramodularity of a certain Calabi-Yau threefold