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.
Cite
@article{arxiv.1004.4699,
title = {Paramodular Abelian Varieties of Odd Conductor},
author = {Armand Brumer and Kenneth Kramer},
journal= {arXiv preprint arXiv:1004.4699},
year = {2018}
}
Comments
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