Let S be the union of all CM-fields and S 0 be the set of non-zero algebraic numbers of S which are not roots of unity. We show that in S 0 Weil's height cannot be bounded from below by an absolute constant.
Let p be a prime number, p ≥ 5. Iwasawa has shown that the p-adic properties of Jacobi sums for Q(ζ p) are linked to Vandiver's Conjecture (see [5]). In this paper, we follow Iwasawa's ideas and study the p-adic properties of the subgroup J of Q(ζ p) * generated by Jacobi sums. Let A be the p-Sylow subgroup of the class group of Q(ζ p). If E denotes the group of units of Q(ζ p), then if Vandiver's Conjecture is true for p, by Kummer theory and class field theory, there is a canonical surjective map Gal(Q(ζ p)(p √ E)/Q(ζ p)) → A − /pA −. Note that J is, for the "minus" part, the analogue of the group of cyclotomic units. We introduce a submodule W of Q(ζ p) * which was already considered by Iwasawa [6]. This module can be thought of, for the minus part, as the analogue of the group of units. We observe that J ⊂ W and if the Iwasawa-Leopoldt Conjecture is true for p then W (Q(ζ p) *) p = J(Q(ζ p) *) p. We prove that if pA − = {0} then (Corollary 4.8) there is a canonical surjective map Gal(Q(ζ p)(p √ W)/Q(ζ p)) → A + /pA + .
Abstract. We give a criterion for the vanishing of the Iwasawa λ-invariants of totally real number fields K based on the class number of K(ζ p ) by evaluating the p-adic L-functions at s = −1.
We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.
For a real abelian number field F and for a prime p we study the relation between the p-parts of the class groups and of the quotients of global units modulo cyclotomic units along the cyclotomic p-extension of F. Assuming Greenberg's conjecture about the vanishing of the λ-invariant of the extension, a map between these groups has been constructed by several authors, and shown to be an isomorphism if p does not split in F. We focus in the split case, showing that there are, in general, non-trivial kernels and cokernels.
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