On the square root of the inverse different

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 Chinburg's second Omega invariant attached to N/F. We show that this equality holds whenever A(N/F) is defined and N/F is tame. This extends a result of the second-named author and S. Vinatier