Paramodular Abelian Varieties of Odd Conductor

Abstract

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.

Frank Calegari was kind enough to point out an oversight in the paramodular conjecture stated in earlier versions. We added Section 8, in which we propose a modified conjecture and show that the modification concerns a rare phenomenon. The numerical evidence in the appendix applies equally to the modified conjecture