Vol. 9, No. 1, 2015

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

Derek Garton

Vol. 9 (2015), No. 1, 149–171

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).

random matrices, function fields, roots of unity, ideal class groups, Cohen–Lenstra heuristics
Mathematical Subject Classification 2010
Primary: 11R29
Secondary: 11R58, 15B52
Received: 26 May 2014
Revised: 18 November 2014
Accepted: 25 December 2014
Published: 18 February 2015
Derek Garton
Fariborz Maseeh Department of Mathematics and Statistics
Portland State University
PO Box 751
Portland, OR 97207-0751
United States