Algebra & Number Theory

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

Derek Garton

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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).

Article information

Algebra Number Theory, Volume 9, Number 1 (2015), 149-171.

Received: 26 May 2014
Revised: 18 November 2014
Accepted: 25 December 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R58: Arithmetic theory of algebraic function fields [See also 14-XX] 15B52: Random matrices

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


Garton, Derek. Random matrices, the Cohen–Lenstra heuristics, and roots of unity. Algebra Number Theory 9 (2015), no. 1, 149--171. doi:10.2140/ant.2015.9.149.

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