Harmonic Analysis and Statistics of the First Galois Cohomology Group

TL;DR

This paper develops harmonic analytic methods to count elements of the first Galois cohomology group with bounded invariants, providing new tools to analyze asymptotic growth and address open questions in number theory.

Contribution

It introduces a harmonic analysis framework using Poisson summation and Euler products to study the distribution of Galois cohomology elements, advancing understanding of Malle's conjecture.

Findings

Established a canonical decomposition of generating series for counting Galois cohomology elements.

Derived asymptotic growth rates using Tauberian theorems.

Resolved some open questions related to the generalized Malle's conjecture.

Abstract

We utilize harmonic analytic tools to count the number of elements of the Galois cohomology group $f\in H^1(K,T)$ with discriminant-like invariant ${\rm inv}(f)\le X$ as $X\to\infty$. Specifically, Poisson summation produces a canonical decomposition for the corresponding generating series as a sum of Euler products for a very general counting problem. This type of decomposition is exactly what is needed to compute asymptotic growth rates using a Tauberian theorem. These new techniques allow for the removal of certain obstructions to known results and answer some outstanding questions on the generalized version of Malle's conjecture for the first Galois cohomology group.