Stark's Conjectures and Hilbert's Twelfth Problem

We give a constructive proof of a theorem given in [Tate 84] which states that (under Stark’s Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real Abelian extension of K. As a direct application of this proof, we show how one can compute explicitly real Abelian extensions of K. We give two examples. In a series of important papers [Stark 71, Stark 75, Stark 76, Stark 80] H. M. Stark developed a body of conjectures relating the values of Artin L-functions at s = 1 (and hence, by the functional equation, their leading terms at s = 0) with certain algebraic quantities attached to extensions of number fields. For example, in the case of Abelian L-functions with a first-order zero at s = 0, the conjectural relation is between the first derivative of the L-functions and the logarithmic embedding of certain units in ray class fields known as Stark units, which are predicted to exist. The use of these conjectures to provide explicit generators of ray class fields, and thus to answer Hilbert’s famous Twelfth Problem was one of the original motivations for their formulation. It has been noticed by several people (including Stark himself [Stark 76]) that they could provide a new way to construct ray class fields of totally real fields. In particular, if K is a totally real field, the field extension generated over K by the Stark units (see below for details) contains the maximal real Abelian extension of K. This result is a direct consequence of Proposition 3.8 (Chap. IV) of [Tate 84]. Using the ideas given in [Stark 76], in Section 2 we give a constructive proof of this result, i.e. for each finite real Abelian extension L/K we construct explicit generators of L over K using Stark units. This proof has a direct application since we can use it to explicitly compute real class fields of a totally real field. This is discussed in Section 3. Since this construction is based on a