AN ALGORITHM FOR COMPUTING THE STICKELBERGER IDEAL FOR MULTIQUADRATIC NUMBER FIELDS

We present an algorithm for computing the Stickelberger ideal for multiquadratic fields K = Q(√d1,√d2, . . . , √dn), where the integers di ≡ 1 mod 4 for i ∈ {1, . . . , n} or dj ≡ 2 mod 8 for one j ∈ {1, . . . , n}; all di’s are pairwise co-prime and squarefree. Our result is based on the paper of Kuˇcera [J. Number Theory, no. 56, 1996]. The algorithm we present works in time O(lg ∆K • 2n• poly(n)), where ∆K is the discriminant of K. As an interesting application, we show a connection between Stickelberger ideal and the class number of a multiquadratic field

An extended Stickelberger ideal of the compositum of a bicyclic field and an imaginary quadratic field

We consider certain extension of the Stickelberger ideal of the compositum of a bicyclic field and a quadratic imaginary field, obtained by adding new annihilators to the Stickelberger ideal. We compute the index of this extension, from which we get some divisibility properties for the relative class number of the compositum.

THE FRACTIONAL GALOIS IDEAL FOR ARBITRARY ORDER OF VANISHING

We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those L-functions of the extension which are non-zero at the special point s = 0, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include L-functions with a simple zero at s = 0, and was shown by the present author to provide class-group annihilators not existing in the Stickelberger ideal. The version presented in this paper deals with L-functions of arbitrary order of vanishing at s = 0, and we give evidence using results of Popescu and Rubin that it is closely related to the Fitting ideal of the class-group, a canonical ideal of annihilators. Finally, we prove an equality involving Stark elements and class-groups originally due to Büyükboduk, but under a slightly different assumption, the advantage being that we need none of the Kolyvagin system machinery used in the original proof.

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

Fix a Galois extension [Formula: see text] 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 [Formula: see text], let [Formula: see text] denote the primes of [Formula: see text] lying above those in S, and let [Formula: see text] denote the ring of [Formula: see text]-integers of [Formula: see text]. We then compare the Fitting ideal of [Formula: see text] 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 [Formula: see text] is a biquadratic extension of F containing the first layer of the cyclotomic ℤ2-extension of F, and describe a class of biquadratic extensions of F = ℚ that satisfy this condition.

Formulae for the relative class number of an imaginary abelian field in the form of a determinant

There is in the literature a lot of determinant formulae involving the relative class number of an imaginary abelian field. Usually such a formula contains a factor which is equal to zero for many fields and so it gives no information about the class number of these fields. The aim of this paper is to show a way of obtaining most of these formulae in a unique fashion, namely by means of the Stickelberger ideal. Moreover some new and non-vanishing formulae are derived by a modification of Ramachandra’s construction of independent cyclotomic units.