Counting number fields using multiple Dirichlet series

TL;DR

This paper introduces a novel analytic method using multiple Dirichlet series to count number fields with fixed Galois groups, providing unconditional results for many groups and conditional asymptotics under subconvexity bounds.

Contribution

It develops a new approach for counting Galois extensions using multiple Dirichlet series, extending results to groups previously inaccessible and including nilpotent groups with power-saving error terms.

Findings

Unconditional results for infinitely many new Galois groups.

Conditional asymptotic growth rates under subconvexity hypotheses.

Power saving error terms for nilpotent groups.

Abstract

We provide a method for counting number fields of fixed Galois group ordered by arbitrary inertial invariants using analytic techniques from the study of multiple Dirichlet series. We prove unconditional results for infinitely many new (concentrated and semiconcentrated) groups that were not approachable by previous methods. Conditional on subconvexity bounds bounds for certain Dirichlet series (e.g. the generalized Lindel\"of hypothesis), we use these techniques to prove the existence of an asymptotic growth rate for $G$-extensions for infinitely many new groups $G$ for which the minimum index elements of $G$ are contained in a union of proper abelian normal subgroups. In particular, our conditional results include all groups with nilpotency class $2$. Additionally, when $G$ is nilpotent our results give a power saving error term.