Values at s = −1 of L -functions for multi-quadratic extensions of number fields and annihilation of the tame kernel

Abstract

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 S E denote the ring of S E -integers of E. We then compare the Fitting ideal of 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.

VALUES AT s = -1 OF L-FUNCTIONS FOR MULTI-QUADRATIC EXTENSIONS OF NUMBER FIELDS, AND THE FITTING IDEAL OF THE TAME KERNEL

“…We state an extended form of it for arbitary finite S which is an easy consequence of the original conjecture for minimal S (see Corollary 3.3 of [11]).…”

“…In the more general setting of a Galois extension of number fields E/F , with Galois group G, Kahn's theorem 5.1 of [6] leads to an exact sequence induced by the transfer map and involving the number r ∞ (E/F ) of infinite primes of F which ramify in E (see [8,Prop. 1.6] or [11,Thm. 3.4]):…”

“…We state a form of it for an arbitrary finite set S which is easily seen to be equivalent to the original conjecture for the minimal choice of the set S, containing just the infinite primes (see Corollary 3.3 of [16]). Deep results on Iwasawa's Main conjecture in [11] and [22] lead to the following (see [8]).…”

L-values and the Fitting ideal of the tame kernel for relative quadratic extensions

“…By Proposition 2.2, the domain and codomain of this map have the same order, and thus the order of the cokernel equals the order of the kernel. The order of ker(ι * ) is computed in [12,Lemma 7.3], yielding 1 if E = F 1 and 2 otherwise.…”

“…We state an extended form of it for arbitrary finite S which is an easy consequence of the original conjecture for minimal S (see Corollary 3.3 of [12]). Deep results on Iwasawa's main conjecture in [10] and [15] lead to the following (see [8]).…”