On the square root of the inverse different

Let F-pi be a finite Galois-algebra extension of a number field F, with group G. Suppose that F-pi/F is weakly ramified and that the square root A(pi) of the inverse different D-pi(-1) is defined. (This latter condition holds if, for example, Gis odd.) Erez has conjectured that the class (A(pi)) of A(pi) in the locally free class group Cl(ZG) of ZG is equal to the Cassou-Nogu & egrave;s-Frohlich root number class W(F-pi/F) associated with F-pi/F. This conjecture has been verified in many cases. We establish a precise formula for (A(pi)) in terms of W(F-pi/F) in all cases where A(pi) is defined and F-pi/F is tame, and are thereby able to deduce that, in general, (A(pi)) is not equal to W(F-pi/F).