Statistics of the First Galois Cohomology Group: A Refinement of Malle's Conjecture

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' counter example to Malle's conjecture and verify the corrected lower bounds predicted by Türkelli.

version 3: A major error is corrected in the proof of the Asymptotic Wiles Theorem after receiving referee feedback, and slightly stronger hypotheses are needed. The main number theoretic results are updated accordingly, namely Theorem 1.4 and Corollary 1.5 have new statements. Other minor corrections and clarifications are included, as well as some improved notation

References in corpus (3)

• Nonabelian Cohen-Lenstra Moments
• Malle's Conjecture for $S_n\times A$ for $n=3,4,5$
• Harmonic Analysis and Statistics of the First Galois Cohomology Group

Cited by in corpus (4)

• Harmonic Analysis and Statistics of the First Galois Cohomology Group
• Malle's conjecture for nonic Heisenberg extensions
• On Malle's conjecture for nilpotent groups, I
• Counting extensions of number fields with Frobenius Galois group