Values at s=-1 of L-functions for multi-quadratic extensions of number fields, and the fitting ideal of the tame kernel

TL;DR

This paper investigates the relationship between the Fitting ideal of K_2 of S-integers in certain Galois extensions and higher Stickelberger ideals, providing new insights under the Birch-Tate conjecture especially for biquadratic extensions.

Contribution

It establishes a comparison between the Fitting ideal and higher Stickelberger ideal in the context of multi-quadratic extensions, extending results to the maximal order of Q[G] and computing indices.

Findings

Fitting and higher Stickelberger ideals coincide in the maximal order of Q[G].

A sufficient condition is identified for the Fitting ideal to contain the higher Stickelberger ideal in biquadratic extensions.

A class of biquadratic extensions satisfying the containment condition is described.

Abstract

Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E lying above those in S, and let O_E^S denote the ring of S_E-integers of E. We then compare the Fitting ideal of K_2(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption of the Birch-Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where E is a biquadratic extension of F containing the first layer of the cyclotomic Z_2-extension of F, and describe a class of biquadratic extensions of F=Q that satisfy this condition.