On the Galois-Gauss sums of weakly ramified characters

TL;DR

This paper investigates the relationship between Galois-Gauss sums and the inverse different in weakly ramified Galois extensions, providing new evidence for a conjecture using algebraic K-theory and prime-power degree extensions.

Contribution

It offers new evidence supporting a conjecture linking Galois-Gauss sums and the inverse different in odd degree extensions, utilizing refined algebraic K-theory techniques.

Findings

Confirmed the conjecture for prime-power degree extensions.

Extended Ullom's result with a refined approach.

Provided concrete examples supporting the theoretical link.

Abstract

Bley, Burns and Hahn used relative algebraic $K$-theory methods to formulate a precise conjectural link 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. We provide concrete new evidence for this conjecture in the setting of extensions of odd prime-power degree by using a refined version of a well-known result of Ullom.