On refined Stark conjectures in the non-abelian case
The conjectures named after Stark come in two types: those ``over \(\mathbb Q\)'' and those ``over \(\mathbb Z\)''. In the former type, very roughly speaking, a regulator and a leading coefficient of an \(L\)-function at \(s=0\) are related by conjecturing (and in many cases proving) that their quotient is algebraic. In the latter, one additionally imposes integrality conditions (bounds on the denominator) on this quotient. In concrete cases this leads to the prediction of the existence of certain units, given by the logarithms of their conjugates. In Stark's seminal work, the setting was nonabelian (so one deals with arbitrary Galois extensions \(K/k\) of number fields), and the conjectures were mostly of the former type. (But in fact Tate's result on rational valued characters specify the rational quotient; these results are used in an essential way in the paper under review.) In later years, conjectures of the latter type were formulated in the abelian case, first for the case where the order of vanishing \(r_\chi\) of the relevant \(L\)-function \(L(s,\chi)\) at \(s=0\) is 1, and then in general (Rubin, Popescu). There is also a contribution of Stark and Chinburg, affording a conjecture over \(\mathbb Z\) in the nonabelian setting, where again \(r_\chi=1\) is assumed. Such a situation arises, by work of Deligne and Serre, from certain modular forms. In the paper under review, the author presents a ``Stark conjecture over \(\mathbb Z\)'' in considerable generality, that is, for any Galois extension \(K/k\) of global fields; the only persistent restriction is that the irreducible character \(\chi\) is assumed to be nontrivial. It is formulated as Conjecture 2.1, which has the form of an inclusion, going from left to right. The term involving the leading coefficient is on the left, and the regulator term on the right. Everything is stated at integral level. Apart from the standard quantities, the left hand side has a factor \(|G|^{r_\chi}\) (with \(G=\text{Gal}(K/k)\)) and the right hand side has a factor \(\text{Fit}(H^{-1}(G,X_S[\chi]))\). (For details we have to refer to the paper.) The author shows in \S3 that up to very minor changes this conjecture specialises to the Stark-Chinburg conjecture. That conjecture has given rise to extremely interesting numerical work; the point is that for \(r_\chi=1\) the conjecture sometimes pins down a conjectural unit so well that one is able to calculate an extremely close approximation to it, hoping to obtain an actual unit. We mention in particular the work of \textit{A. Jehanne, J. Sands} and \textit{X.-F. Roblot} [Exp. Math. 12, No. 4, 419--432 (2003; Zbl 1082.11082)] on icosahedral representations \(\chi\). The paper under review points out a (seemingly harmless but not negligible) oversight in that paper: due to a missing exponent 2 several statements have to be recast (details: see remark 3.4), but the substance, and the main result in particular, is not at all affected. In the final \S4 of the paper under review, the author gives a proof of his Conjecture in quite a few cases. Apart from a certain technical step (reduction to ``large'' \(S\), for the experts), the main point is a reduction to the so-called Strong Stark Conjecture [see \textit{T. Chinburg}, ``On the Galois structure of algebraic integers and S-units'', Invent. Math. 74, 321--349 (1983; Zbl 0564.12016)], and this conjecture is already known to hold in a variety of cases. This includes the function field case, rational characters (any number field \(k\)), one-dimensional characters (\(k\) the rationals, or \(k\) imaginary quadratic with extra conditions). For all these cases, and of course all other number field extensions \(K/k\) for which the Strong Stark Conjecture is true, the conjecture of the paper under review is therefore established.
scientific article; zbMATH DE number 5505923