#### Vol. 15, No. 10, 2021

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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 $N\left(K,G;X\right)$, 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 ${Z}^{1}\left(K,G\right)$ (or equivalently $1$-coclasses in ${H}^{1}\left(K,G\right)$) 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 ${Z}^{1}\left(K,T\right)$ (and equivalently for ${H}^{1}\left(K,T\right)$) 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 $N\left(K,G;X\right)$ 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.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/ant

We have not been able to recognize your IP address 3.215.79.204 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.