On the square root of the inverse different

TL;DR

This paper investigates the relationship between the class of the square root of the inverse different in Galois extensions of number fields and the Cassou-Nogues-Frohlich root number class, providing a formula and counterexamples.

Contribution

It establishes a precise formula for the difference between the class of A(N/F) and W(N/F) in tame extensions, showing they are not always equal.

Findings

Derived a formula for (A(N/F)) - W(N/F) in tame cases

Showed (A(N/F)) is not always equal to W(N/F)

Connected class differences to Galois-Gauss sum signs

Abstract

Let N/F be a finite, normal extension of number fields with Galois group G. Suppose that N/F is weakly ramified, and that the square root A(N/F) of the inverse different of N.F is defined. (This latter condition holds if, for example, G is of odd order.) B. Erez has conjectured that the class (A(N/F)) of A(N/F) in the locally free class group Cl(ZG) of ZG is equal to the Cassou-Nogues-Frohlich root number class W(N/F) attached to N/F. We establish a precise formula for (A(N/F)) - W(N/F) in terms of the signs of certain symplectic Galois-Gauss sums whenever N/F is tame and (A(N/F)) is defined. We thereby show that, in general, (A(N/F)) is not equal to W(N/F).