Random matrices, the Cohen–Lenstra heuristics, and roots of unity

Garton, Derek

doi:10.2140/ant.2015.9.149

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Random matrices, the Cohen–Lenstra heuristics, and roots of unity

Derek Garton

Vol. 9 (2015), No. 1, 149–171

DOI: 10.2140/ant.2015.9.149

Abstract

The Cohen–Lenstra–Martinet heuristics predict the frequency with which a fixed finite abelian group appears as an ideal class group of an extension of number fields, for certain sets of extensions of a base field. Recently, Malle found numerical evidence suggesting that their proposed frequency is incorrect when there are unexpected roots of unity in the base field of these extensions. Moreover, Malle proposed a new frequency, which is a much better match for his data. We present a random matrix heuristic (coming from function fields) that leads to a function field version of Malle’s conjecture (as well as generalizations of it).

Keywords

random matrices, function fields, roots of unity, ideal class groups, Cohen–Lenstra heuristics

Mathematical Subject Classification 2010

Primary: 11R29

Secondary: 11R58, 15B52

Milestones

Received: 26 May 2014

Revised: 18 November 2014

Accepted: 25 December 2014

Published: 18 February 2015

Authors

Derek Garton
Fariborz Maseeh Department of Mathematics and Statistics Portland State University PO Box 751 Portland, OR 97207-0751 United States

Vol. 9, No. 1, 2015