Vol. 12, No. 5, 2018

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Certain abelian varieties bad at only one prime

Armand Brumer and Kenneth Kramer

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

An abelian surface ${A}_{∕ℚ}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathsc{S}}_{5}$-extension over $ℚ$ with ramification index 5 over ${ℚ}_{2}$. Let $A$ be favorable and let $B$ be a semistable abelian variety of dimension $2d$ and conductor ${N}^{d}$ with $B\left[2\right]$ filtered by copies of $A\left[2\right]$. We give a sufficient class field theoretic criterion on $F$ to guarantee that $B$ is isogenous to ${A}^{d}$.

As expected from our paramodular conjecture, we conclude that there is one isogeny class of abelian surfaces for each conductor in $\left\{277,349,461,797,971\right\}$. 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