Function fields with isomorphic Galois groups

Abstract:

Let K be a local field or a global field of characteristic p. Let ${G_K}$ be the Galois group of the separable closure of K over K. In the local case we show that ${G_K}$, considered as an abstract profinite group, determines the characteristic of K and the number of elements in the residue class field. In the global case we show that ${G_K}$ determines the number of elements in the constant field of K as well as the zeta function, genus and class number of K. Let $K’$ be another global field of characteristic p and assume we have $\lambda :{G_K} \to {G_{K’}}$, an isomorphism of profinite groups. Then K and $K’$ have the same constant field, zeta function, genus and class number. We also prove that the idele class groups and divisor class groups of K and $K’$ are isomorphic. If E is a finite extension of k, the constant field of K and $K’$, we show that the E-rational points of the Jacobian varieties of K and $K’$ are isomorphic as $G(E/k)$-modules. If $K = K’$ and $\bar K = \bar kK$ where $\bar k$ is the algebraic closure of k, we prove that $\lambda ({G_{\bar K}}) = {G_{\bar K}}$ and the induced automorphism of $G(\bar K/K)$ is the identity.