Arithmetic statistics of $H^1(K, T)$

Abstract: Coclasses in a Galois cohomology group $H^1(K, T)$ parametrize extensions of a number field with certain Galois group. It is natural to want to count these coclasses with general local conditions and with respect to a discriminant-like invariant. In joint work with Brandon Alberts, I present a novel tool for studying this: harmonic analysis on adelic cohomology, modeled after the celebrated use of harmonic analysis on the adeles in Tate's thesis. This leads to a more illuminating explanation of a fact previously noticed by Alberts, namely that the Dirichlet series counting such coclasses is a finite sum of Euler products; and we nail down the asymptotic count of coclasses in satisfying generality.

number theory

Audience: researchers in the topic

( paper )

Comments: In the pre-talk, I will give a rundown on the needed background in Galois cohomology, etale algebras, the local Tate pairing, and Poitou-Tate duality.

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UCSD number theory seminar

Series comments: Most talks are preceded by a pre-talk for graduate students and postdocs. The pre-talks start 40 minutes prior to the posted time (usually at 1:20pm Pacific) and last about 30 minutes.

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