Distributions of unramified extensions of global fields

Abstract: Every number field K has a maximal unramified extension K^un, with Galois group Gal(K^un/K) (whose abelianization is the class group of K).  As K varies, we ask about the distribution of the groups Gal(K^un/K).  We prove some results about the structure of Gal(K^un/K) that motivate us to give a conjecture about this distribution, which we also conjecture in the function field analog.  We give theorems in the function field case (as the size of the finite field goes to infinity) that support these new conjectures.  In particular, our distributions abelianize to the Cohen-Lenstra-Martinet distributions for class groups, and so our function field theorems prove (suitably modified) versions of the Cohen-Lenstra-Martinet heuristics over function fields as the size of the finite field goes to infinity.  This talk is on joint work with Yuan Liu and David Zureick-Brown.

algebraic geometrynumber theory

Audience: researchers in the topic

( paper )

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MIT number theory seminar

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