Elsevier

Journal of Number Theory

Volume 218, January 2021, Pages 161-179
Journal of Number Theory

General Section
A genus formula for the positive étale wild kernel

https://doi.org/10.1016/j.jnt.2020.07.013Get rights and content

Abstract

Let F be a number field and let i2 be an integer. In this paper, we study the positive étale wild kernel WK2i2ét,+F, which is the twisted analogue of the 2-primary part of the narrow class group. If E/F is a Galois extension of number fields with Galois group G, we prove a genus formula relating the order of the groups (WK2i2ét,+E)G and WK2i2ét,+F.

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 GF,S be the Galois group of the maximal algebraic extension FS of F which is unramified outside S. It is well known that for all integer i2, 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 WK2i2étF. It is the analog of the p-part of the classical wild kernel WK2F which occurs in Moore's exact sequence of power norm symbols (cf. [17]).

Let E/F 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 p=2. 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 Sf denotes the set of finite primes in S and H+j(.,.) denotes the j-th totally positive Galois cohomology groups (Section 2.1). When i=1, this kernel is isomorphic to the 2-primary part of the narrow S-class group of F. For i2, 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 WK2i2ét,+F. It is analogue to the narrow S-class group of F, and fits into an exact sequence (Proposition 2.4) where DF(i) is the étale Tate kernel and DF+(i)/F2 is the kernel of the signature map ([13, Definition 2.4]). Here r1 denotes the number of real places of F. In particular,

  • for i even, we haveWK2i2ét,+FWK2i2étF.

  • for i odd, we have an exact sequence where δi(F) is the 2-rank of the cokernel of the signature map sgnF.

For a place v of F, let Gv denote the decomposition group of v in E/F. Following [3, Definition 2.6], define the plus normic subgroup H+1,N(F,Z2(i)) to be the kernel of the map where NGv=σGvσ is the norm map, and if v is a prime of F, we denote by w a prime of E above v. We prove the following genus formula for the positive étale wild kernel.

Theorem

Let E/F be a Galois extension of number fields with Galois group G. Then for every i2, we have|(WK2i2ét,+E)G||WK2i2ét,+F|=|XE/F(i)|.vSf|H1(Gv,H2(Ew,Z2(i)))||H1(G,H0(E,Q2/Z2(1i)))|.[H+1(F,Z2(i)):H+1,N(F,Z2(i))].

The group XE/F(i) (Definition 3.4) has order at most |H2(G,H0(E,Q2/Z2(1i)))|, and is trivial if the canonical morphism is surjective. In particular, if E/F is a relative quadratic extension of number fields, the order of the group XE/F(i) is at most 2. In this case we give, in the last section, a genus formula involving the positive Tate kernel DF+(i). Roughly speaking, let F (resp. Fv,) denote the cyclotomic Z2-extension of F (resp. Fv) and let RE/F be the set of both finite primes tamely ramified in E/F and 2-adic primes such that EwFv,. Then for any odd integer i2, we have
  • (i)

    if EF, the positive étale wild kernel satisfies Galois codescent;

  • (ii)

    if EF,|(WK2i2ét,+E)G||WK2i2ét,+F|=2r(E/F)1+t[DF+(i):DF+(i)NGE] where r(E/F)=|RE/F| and t{0,1}. Moreover, t=0 if RE/F.

Section snippets

Totally positive Galois cohomology

If K is a field, Ksep will denote a fixed separable closure of K and GK=Gal(Ksep/K). Let F be a number field, we write S for a finite set of primes of F containing the set S2 of dyadic primes and the set S of archimedean primes. For a place v of F, we denote by Fv the completion of F at v. Note that for any infinite place v of F, a fixed embedding FsepFvsep extending the embedding FFv defines a continuous homomorphism

For a discrete or a compact Z2[[GF,S]]-module M, we write M+ for

Genus formula

Let E/F 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 E/F. We denote also by S the set of places of E above places in S. In the sequel we assume that i2. By (4), we have the exact sequence where ˜vSfH2(Fv,Z2(i)) 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 E/F with Galois group G. For such extensions, we give a genus formula for the positive étale wild kernel involving the norm index [DF+(i):DF+(i)NGE], where DF+(i) is the positive Tate kernel. First recall that for every even integer i2, 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 i=2, and by

Acknowledgment

We would like to thank the anonymous referee whose comments and suggestions helped improve this paper.

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