Paramodular Abelian Varieties of Odd Conductor

Details Paramodular Abelian Varieties of Odd Conductor Paramodular Abelian Varieties of Odd Conductor

2010-04-27

arxiv.org/abs/1004.4699v2

Mathematics

Number Theory

Additional comments on Weil restrictions and reference added.

Scientific paper

A precise and testable modularity conjecture for rational abelian surfaces A with trivial endomorphisms, End_Q A = Z, is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen on weight 2 Siegel paramodular forms. We obtain fairly precise information on ell-division fields of semistable abelian varieties A, mainly when A[ell] is reducible, by considering extension problems for groups schemes of small rank. Our general results imply, for instance, that the least prime conductor of an abelian surface is 277.

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