Algebra & Number Theory

Certain abelian varieties bad at only one prime

Armand Brumer and Kenneth Kramer

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An abelian surface A of prime conductor N is favorable if its 2-division field F is an S 5 -extension over with ramification index 5 over 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 { 2 7 7 , 3 4 9 , 4 6 1 , 7 9 7 , 9 7 1 } . The general applicability of our criterion is discussed in the data section.

Article information

Algebra Number Theory, Volume 12, Number 5 (2018), 1027-1071.

Received: 1 September 2016
Revised: 20 August 2017
Accepted: 23 October 2017
First available in Project Euclid: 14 August 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx]
Secondary: 11R37: Class field theory 11S31: Class field theory; $p$-adic formal groups [See also 14L05] 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]

semistable abelian variety group scheme Honda system conductor paramodular conjecture


Brumer, Armand; Kramer, Kenneth. Certain abelian varieties bad at only one prime. Algebra Number Theory 12 (2018), no. 5, 1027--1071. doi:10.2140/ant.2018.12.1027.

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