Distributions of unramified extensions of global fields

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• Melanie Matchett Wood (Harvard)
• Wednesday 10 May 2023, 14:30-15:30
• MR15.

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Every number field K has a maximal unramified extension Kun, with Galois group Gal(Kun/K) (whose abelianization is the class group of K). As K varies, we ask about the distribution of the groups Gal(Kun/K). We give a conjecture about this distribution, which we also conjecture in the function field analog. We give some results about Gal(Kun/K) that motivate us to build certain random groups whose distributions appear in our conjectures. 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 give support to (suitably modified) versions of the Cohen-Lenstra-Martinet heuristics. This talk is on joint work with Yuan Liu and David Zureick-Brown, and with Will Sawin.

This talk is part of the Number Theory Seminar series.

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