Washington units, semispecial units, and annihilation of class groups

Special units are a sort of predecessor of Euler systems, and they are mainly used to obtain annihilators for class groups. So one is interested in finding as many special units as possible (actually we use a technical generalization called “semispecial”). In this paper we show that in any abelian field having a real genus field in the narrow sense all Washington units are semispecial, and that a slightly weaker statement holds true for all abelian fields. The group of Washington units is very often larger than Sinnott’s group of cyclotomic units. In a companion paper we will show that in concrete families of abelian fields the group of Washington units is much larger than that of Sinnott units, by giving lower bounds on the index. Combining this with the present paper gives strong annihilation results.

Fermat curves and a refinement of the reciprocity law on cyclotomic units

We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

On an isomorphism lying behind the class number formula

Let p be an odd prime such that the Greenberg conjecture holds for the maximal real cyclotomic subfield K1 of Q[ζp]. Let An = (C(Kn))p be the p-part of the class group of Kn, the n-th field in the cyclotomic tower, and let En, Cn be the global and cyclotomic units of Kn, respectively. We prove that under this premise, there is some n0 such that for all m ≥ n0, the class number formula (Em/Cm)p = |Am| hides in fact an isomorphism of Λ[Gal(K1/Q)]-modules.

Cyclotomic units and class groups in p-extensions of real abelian fields

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.