On Galois-Gauss sums and the square root of the inverse different

TL;DR

This paper explores a generalization of a conjecture relating Galois-Gauss sums and the square root of the inverse different in number field extensions, extending previous results to broader classes of Galois extensions.

Contribution

It extends the conjecture and methods of Bley, Burns, and Hahn to all finite Galois extensions with a square root of the inverse different, incorporating recent results on Artin root numbers.

Findings

Extended the conjecture to all relevant Galois extensions.

Provided new insights into Erez's conjecture on Galois module structure.

Unified methods for analyzing Galois-Gauss sums and inverse differents.

Abstract

We discuss a possible generalisation of a conjecture of Bley, Burns and Hahn concerning the relation between the second Adams-operator twisted Galois-Gauss sums of weakly ramified Artin characters and the square root of the inverse different of finite, odd degree, Galois extensions of number fields, to the setting of all finite Galois extensions of number fields for which a square root of the inverse different exists. We also extend the key methods and results of Bley, Burns and Hahn to this more general setting and, by combining these methods with a recent result of Agboola, Burns, Caputo and the present author concerning Artin root numbers of twisted irreducible symplectic characters, we provide new insight into a conjecture of Erez concerning the Galois structure of the square root of the inverse different.