Thursday 31 Oct 2013Multiplicative Galois module structure and the Cohen-Lenstra heuristics

Alex Bartel - University of Warwick

Harrison 004 15:00-16:00

If F/K is a Galois extension of number fields, then the integral units of F modulo the roots of unity form a Z-free Galois module, and it is a classical problem to understand the structure of this module. A natural question that seems to have received surprisingly little attention is: as F ranges over Galois extensions of K with a fixed Galois group G, how often are the (finitely many) possible integral representations of G realised as the integral units of F modulo roots of unity? I will explain in detail the problem, giving concrete examples, and will then propose a prediction of these frequencies, which not only deals with the Galois module structure of the units, but also contains the classical Cohen-Lenstra heuristic on distributions of class groups as a special case.

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