Statistics of the first Galois cohomology group : A refinement of Malle’s conjecture

Alberts, Brandon

doi:10.2140/ant.2021.15.2513

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Statistics of the first Galois cohomology group: A refinement of Malle's conjecture

Brandon Alberts

Vol. 15 (2021), No. 10, 2513–2569

DOI: 10.2140/ant.2021.15.2513

Abstract

Malle proposed a conjecture for counting the number of $G$ -extensions $L ∕ K$ with discriminant bounded above by $X$ , denoted $N (K, G; X)$ , where $G$ is a fixed transitive subgroup $G \subset S_{n}$ and $X$ tends towards infinity. We introduce a refinement of Malle’s conjecture, if $G$ is a group with a nontrivial Galois action then we consider the set of crossed homomorphisms in $Z^{1} (K, G)$ (or equivalently $1$ -coclasses in $H^{1} (K, G)$ ) with bounded discriminant. This has a natural interpretation given by counting $G$ -extensions $F ∕ L$ for some fixed $L$ and prescribed extension class $F ∕ L ∕ K$ .

If $T$ is an abelian group with any Galois action, we compute the asymptotic growth rate of this refined counting function for $Z^{1} (K, T)$ (and equivalently for $H^{1} (K, T)$ ) and show that it is a natural generalization of Malle’s conjecture. The proof technique is in essence an application of a theorem of Wiles on generalized Selmer groups, and additionally gives the asymptotic main term when restricted to certain local behaviors. As a consequence, whenever the inverse Galois problem is solved for $G \subset S_{n}$ over $K$ and $G$ has an abelian normal subgroup $T ⊴ G$ we prove a nontrivial lower bound for $N (K, G; X)$ given by a nonzero power of $X$ times a power of $log X$ . For many groups, including many solvable groups, these are the first known nontrivial lower bounds. These bounds prove Malle’s predicted lower bounds for a large family of groups, and for an infinite subfamily they generalize Klüners’ counterexample to Malle’s conjecture and verify the corrected lower bounds predicted by Türkelli.

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Keywords

Malle's conjecture, arithmetic statistics, first cohomology group, crossed homomorphisms, Malle–Bhargava principle, Wiles' theorem

Mathematical Subject Classification

Primary: 11R21

Secondary: 11R32, 11R34

Milestones

Received: 19 February 2020

Revised: 19 January 2021

Accepted: 24 February 2021

Published: 8 February 2022

Authors

Brandon Alberts
Department of Mathematics University of California San Diego La Jolla, CA United States

Vol. 15, No. 10, 2021