General SectionA genus formula for the positive étale wild kernel
Introduction
Let F be a number field and let p be a prime number. For a finite set S of primes of F containing the p-adic primes and the infinite primes, let be the Galois group of the maximal algebraic extension of F which is unramified outside S. It is well known that for all integer , the kernel of the localization map is independent of the choice of the set S [19, §6, Lemma 1] [13, page 336]. This kernel is called the étale wild kernel [6], [12], and denoted by . It is the analog of the p-part of the classical wild kernel which occurs in Moore's exact sequence of power norm symbols (cf. [17]).
Let be a Galois extension with Galois group G. For a fixed odd prime p, several authors have studied Galois co-descent and proved genus formulas for the étale wild kernel [14], [9], [2], [1], which are analogues to the Chevalley genus formula for the class groups. In this paper we settle the case . For this purpose, we use a slight variant of cohomology, the so-called totally positive Galois cohomology [10, §5] [5, §2]. More precisely, for all integer i, we are interested in the kernel of the map Here denotes the set of finite primes in S and denotes the j-th totally positive Galois cohomology groups (Section 2.1). When , this kernel is isomorphic to the 2-primary part of the narrow S-class group of F. For , we show that this kernel is independent of the set S; it is referred to as the positive étale wild kernel, and is denoted by . It is analogue to the narrow S-class group of F, and fits into an exact sequence (Proposition 2.4) where is the étale Tate kernel and is the kernel of the signature map ([13, Definition 2.4]). Here denotes the number of real places of F. In particular,
- •
for i even, we have
- •
for i odd, we have an exact sequence where is the 2-rank of the cokernel of the signature map .
Theorem
Let be a Galois extension of number fields with Galois group G. Then for every , we have
- (i)
if , the positive étale wild kernel satisfies Galois codescent;
- (ii)
if , where and . Moreover, if .
Section snippets
Totally positive Galois cohomology
If K is a field, will denote a fixed separable closure of K and . Let F be a number field, we write S for a finite set of primes of F containing the set of dyadic primes and the set of archimedean primes. For a place v of F, we denote by the completion of F at v. Note that for any infinite place v of F, a fixed embedding extending the embedding defines a continuous homomorphism For a discrete or a compact -module M, we write for
Genus formula
Let be a Galois extension of number fields with Galois group G. Let S denote the set of infinite places, 2-adic places and those which ramify in . We denote also by S the set of places of E above places in S. In the sequel we assume that . By (4), we have the exact sequence where denotes the kernel of the surjective map Then the corestriction map induces the exact commutative diagram where the middle vertical map is an isomorphism by (3). Using the snake lemma, we
Relative quadratic extension case
In this section we focus on relative quadratic extensions of number fields with Galois group G. For such extensions, we give a genus formula for the positive étale wild kernel involving the norm index , where is the positive Tate kernel. First recall that for every even integer , Proposition 2.4 says that the étale wild kernel and the positive étale wild kernel coincide. A genus formula has been obtained by Kolster-Movahhedi [14, Theorem 2.18] for , and by
Acknowledgment
We would like to thank the anonymous referee whose comments and suggestions helped improve this paper.
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