Galois co-descent for étale wild kernels and capitulation

Abstract

Let F be a number field with ring of integers o F . For a fixed prime number p and i≥2 the étale wild kernels WK 2i-2 e ´t (F) are defined as kernels of certain localization maps on the i-fold twist of the p-adic étale cohomology groups of spec o F [1 p]. These groups are finite and coincide for i=2 with the p-part of the classical wild kernel WK 2 (F). They play a role similar to the p-part of the p-class group of F. For class groups, Galois co-descent in a cyclic extension L/F is described by the ambiguous class formula given by genus theory. In this formula, the only factor which is not well mastered is the norm index [U F ′ :U F ′ ∩N L/F (L * )] for the p-units U F ′ . The aim of this paper is the study of the Galois co-descent for wild kernels: Given a cyclic extension L/F of degree p with Galois group G, we show that the transfer map WK 2i-2 e ´t (L) G →WK 2i-2 e ´t (F) is onto except in a very special case, then we determine its kernel as the cokernel of a certain cup-product with values in a Brauer group. This approach also yields a genus formula, analogous to the one for class groups, comparing the sizes of WK 2i-2 e ´t (L) G and WK 2i-2 e ´t (F) where p-units U F ′ are replaced by odd K-theory groups. When p is odd, we illustrate the method by finding all Galois p-extensions of Q, for which the p-part of the classical wild kernel is trivial. For p≥5,they turn out to be the layers of the cyclotomic Z p -extension of Q.