Divisibility properties of values of partial zeta functions at non-positive integers
To an abelian extension \(K/k\) of number fields with group \(G\) one may attach an equivariant (imprimitive) \(L\)-value \(\theta_{K/k}\) at \(s=0\). This generalised Stickelberger element is an element of the group ring \(\mathbb Q[G]\). It may have denominators, but they are bounded by \(w\), the number of roots of unity in \(K\). If \(I\) is the annihilator of \(\mu_K\) in \(Z[G]\), then \(I\theta_{K/k}\) is conjectured to annihilate the class group (Brumer's conjecture); there is an analog for higher K-groups (Coates-Sinnott conjecture) where \(s=0\) is replaced by \(s=-n\), a negative integer. It is a natural question what the denominator really is; more precisely, what the denominators of the coefficients are, when \(\theta_{K/k}\) is written in the form \(\sum_{\sigma\in G}a_\sigma\,\sigma\) with rational \(a_\sigma\). \textit{D. R. Hayes} has observed [Burns, David (ed.) et al., Stark's conjectures: recent work and new directions, Contemp. Math. 358, 193--206 (2004; Zbl 1076.11057)] that all \(a_\sigma\) have the same denominator (assuming of course that we write them in lowest terms). He also formulated conjectures concerning these denominators. The author of the paper under review formulates a closely related one, called Conjecture 1: Take \(s=0\). If \(w=|\mu_K|\) is exactly divisible by \(p^r\), \(\mu_K\) is \(G\)-cohomologically trivial, and there exists an unramified cyclic degree \(p^r\) extension \(L/K\) such that the associated Kummer radical is fixed under \(G\), then \(\theta_{K/k}\) has no \(p\) in its denominators at all. There are two more conjectures, one of them bearing on the case \(s=-n\). The author indicates why these conjectures are close to those of Hayes (which we do not formulate here for lack of space), and he proceeds to show that the Coates-Sinnott conjecture implies Conjecture 2, and that the Brumer conjecture implies Conjecture 1 and 3. In a final section he discusses the ``dynamics'' of his conjectures, that is, their behaviour when the initial data change, for example what happens under change of fields. Small remark of the reviewer: In correspondence with the author the fact has emerged that \(r=1\) cannot happen in the setting of Conjecture 1 as soon as the order of \(G\) is divisible by \(p\). So the realm of interest of this conjecture is bipartite: either \(p\) does not divide \(|G|\) (perhaps a somewhat simpler situation), or \(|G|\) is a multiple of \(p\) and \(r\) is at least 2. The paper is accessible to non-specialists, since the author takes great care in setting the scene and reviewing the necessary concepts in his Introduction.
scientific article; zbMATH DE number 6031803