Paramodular Abelian Varieties of Odd Conductor

TL;DR

This paper proposes a modularity conjecture for rational abelian surfaces with trivial endomorphisms, providing insights into their division fields and establishing bounds on prime conductors.

Contribution

It introduces a new modularity conjecture for abelian surfaces and derives results on division fields and prime conductors, supported by examples and recent research.

Findings

Least prime conductor of an abelian surface is 277

Provides detailed information on ell-division fields for semistable abelian varieties

Supports conjecture with examples and non-existence results

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.