Vol. 15, No. 10, 2021

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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 LK with discriminant bounded above by X, denoted N(K,G;X), where G is a fixed transitive subgroup G Sn 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 Z1(K,G) (or equivalently 1-coclasses in H1(K,G)) with bounded discriminant. This has a natural interpretation given by counting G-extensions FL for some fixed L and prescribed extension class FLK.

If T is an abelian group with any Galois action, we compute the asymptotic growth rate of this refined counting function for Z1(K,T) (and equivalently for H1(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 Sn 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 logX. 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