On the square root of the inverse different

Agboola, Adebisi  ; Burns, David John ; Caputo, Luca ; Kuang, Yu

Canadian journal of mathematics, Tome 76 (2024) no. 1, pp. 283-318

Résumé

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 $\mathfrak {D}_{\pi }^{-1}$ is defined. (This latter condition holds if, for example, $|G|$ is odd.) Erez has conjectured that the class $(A_\pi )$ of $A_\pi $ in the locally free class group $\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ of $\mathbf {Z} G$ is equal to the Cassou–Noguès–Fröhlich 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 $W(F_\pi /F)$.

Détail

DOI : 10.4153/S0008414X23000019

Mots-clés : Square root of inverse different, Galois module structure, Galois–Jacobi sum, Artin root number