Malle's conjecture with multiple invariants

TL;DR

This paper introduces multiple invariants for Galois extensions of number fields, proposes a heuristic generalizing Malle's conjecture, and proves it for abelian groups, also relating invariants to refined Artin conductors.

Contribution

It defines new invariants for Galois extensions, formulates a multi-invariant heuristic version of Malle's conjecture, and proves this conjecture in the abelian case.

Findings

Heuristic predicts the distribution of extensions with multiple invariants.

Conjecture is proved for abelian Galois groups.

Refined Artin conductors encode the same information as the invariants.

Abstract

We define invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$ of Galois extensions of number fields with a fixed Galois group. Then, we propose a heuristic in the spirit of Malle's conjecture which asymptotically predicts the number of extensions that satisfy $\operatorname{inv}_i\leq X_i$ for all $X_i$. The resulting conjecture is proved for abelian Galois groups. We also describe refined Artin conductors that carry essentially the same information as the invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$.