Washington units, semispecial units, and annihilation of class groups
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© The Author(s) 2020
Cornelius Greither · Radan Kuˇcera
Washington units, semispecial units, and annihilation of class groups Received: 28 November 2019 / Accepted: 19 August 2020 Abstract. 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.
By an abelian field, we will always mean a finite abelian extension K of Q. The first aim of this note is to show that under a suitable definition of semispecialness all Washington units of an abelian field with real genus field (in the narrow sense) are semispecial. Washington units are a certain kind of circular units; for the precise definition see below. This occupies the first section. In the second section, we prove a somewhat weaker result for abelian fields whose genus field is imaginary; this will not be used in the sequel. The main reason why one is interested in semispecial numbers or units in an abelian field K is that they lead to annihilation statements for the class group of K . Semispecial units are a generalization of Rubin’s notion of “special units” (see [7]), and these special units were used by Rubin, based on pioneering work of Thaine, to obtain annihilation statements for class groups. Shortly after that, the theory of Euler systems was developed, which has led to spectacular results about the structure of class groups and Main Conjectures, but we will not be concerned with Euler systems here. It has turned out that in some situations, special units are hard to come by, so we were led to the notion of “semispecial units”, which is maybe less pleasant to the eye but more pliable, and still can be used to obtain annihilation results for class groups. In the last section of this note, we will explain why an annihilation result of this kind that was proved in earlier work of the authors C. Greither (B): UniBw München, 85577 Neubiberg, Germany. e-mail: [email protected] R. Kuˇcera: Masaryk University, 611 37 Brno, Czech Republic. e-mail: [email protected] Mathematics Subject Classification: 11R20
https://doi.org/10.1007/s00229-020-01241-y
C. Greither, R. Kuˇcera
for a very restricted class of real abelian fields remains valid for many more abelian fields. The immediate purpose of this is to lay the ground for another paper of ours, in which we treat a particular class of abelian fields K with
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