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

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.

Malle's conjecture, arithmetic statistics, first cohomology group, crossed homomorphisms, Malle–Bhargava principle, Wiles' theorem
Mathematical Subject Classification
Primary: 11R21
Secondary: 11R32, 11R34
Received: 19 February 2020
Revised: 19 January 2021
Accepted: 24 February 2021
Published: 8 February 2022
Brandon Alberts
Department of Mathematics
University of California San Diego
La Jolla, CA
United States