##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Evan O'Dorney

#### Princeton University

## 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.

Host: Kiran Kedlaya

### May 27, 2021

### 2:00 PM

### See https://www.math.ucsd.edu/\~{}nts/

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