On Poincaré sums for local fields

Abstract

Let $K/k$ be a finite Galois extension of local fields. To each class $\gamma = [c]$ in $H^1(\operatorname{Gal}(K/k), U_K)$, $U_K$ being the group of units of $K$, we associate an index $i_\gamma(K/k) = (M_c : P_c)$ after the model of Poincaré series and study its relation to the ramification theory of $K/k$.