Function fields with isomorphic Galois groups

Abstract

Let K be a local field or a global field of characteristic p. Let G K {G_K} be the Galois group of the separable closure of K over K. In the local case we show that G K {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 {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 ′ K’ be another global field of characteristic p and assume we have λ : G K → G K ′ \lambda :{G_K} \to {G_{K’}} , an isomorphism of profinite groups. Then K and K ′ 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 ′ K’ are isomorphic. If E is a finite extension of k, the constant field of K and K ′ K’ , we show that the E-rational points of the Jacobian varieties of K and K ′ K’ are isomorphic as G ( E / k ) G(E/k) -modules. If K = K ′ K = K’ and K ¯ = k ¯ K \bar K = \bar kK where k ¯ \bar k is the algebraic closure of k, we prove that λ ( G K ¯ ) = G K ¯ \lambda ({G_{\bar K}}) = {G_{\bar K}} and the induced automorphism of G ( K ¯ / K ) G(\bar K/K) is the identity.