Vol. 12, No. 5, 2018

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Certain abelian varieties bad at only one prime

Armand Brumer and Kenneth Kramer

Vol. 12 (2018), No. 5, 1027–1071
Abstract

An abelian surface A of prime conductor N is favorable if its 2-division field F is an S5-extension over with ramification index 5 over 2. Let A be favorable and let B be a semistable abelian variety of dimension 2d and conductor Nd with B[2] filtered by copies of A[2]. We give a sufficient class field theoretic criterion on F to guarantee that B is isogenous to Ad.

As expected from our paramodular conjecture, we conclude that there is one isogeny class of abelian surfaces for each conductor in {277,349,461,797,971}. The general applicability of our criterion is discussed in the data section.

Keywords
semistable abelian variety, group scheme, Honda system, conductor, paramodular conjecture
Mathematical Subject Classification 2010
Primary: 11G10
Secondary: 11R37, 11S31, 14K15
Milestones
Received: 1 September 2016
Revised: 20 August 2017
Accepted: 23 October 2017
Published: 31 July 2018
Authors
Armand Brumer
Department of Mathematics
Fordham University
Bronx, NY
United States
Kenneth Kramer
Department of Mathematics
Queens College (CUNY)
Flushing, NY
United States