Certain abelian varieties bad at only one prime

Brumer, Armand; Kramer, Kenneth

doi:10.2140/ant.2018.12.1027

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Certain abelian varieties bad at only one prime

Armand Brumer and Kenneth Kramer

Vol. 12 (2018), No. 5, 1027–1071

DOI: 10.2140/ant.2018.12.1027

Abstract

An abelian surface $A_{∕ ℚ}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an $S_{5}$ -extension over $ℚ$ with ramification index 5 over $ℚ_{2}$ . Let $A$ be favorable and let $B$ be a semistable abelian variety of dimension $2 d$ and conductor $N^{d}$ 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 $A^{d}$ .

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.

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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

Vol. 12, No. 5, 2018