Paramodular Abelian Varieties of Odd Conductor

2010/04/27 by Brumer, Armand, Kramer, Kenneth · 1 citation
#11F46 (Secondary) #11G10 (Primary) 14K15 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1004.4699

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.

Cited by

• Modularity theorems for abelian surfaces

Related

• Arithmetic of Division Fields
• Paramodular Cusp Forms
• Modular Abelian Varieties of Odd Modular Degree
• Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups
• Antisymmetric Paramodular Forms of Weights 2 and 3