Stickelberger elements over rational function fields
For an abelian extension \(L/K\) of global fields, Gross has conjectured a refined analytic class number formula relating the Stickelberger element \(\theta\) of the extension to a certain refined regulator \(\mathcal{R}\). The refined regulator, defined as an element of \(I^n/I^{n+1}\) where \(I\) is the augmentation ideal \(I\) of \(G = \mathrm{Gal}(L/K)\) and \(n\) is a certain integer, is given in terms of local reciprocity maps. To explain the integer \(n\), we remark that the setup requires one to choose a finite set \(S\) of places of \(K\) containing the infinite and ramified ones and a finite set \(T\) of places of \(K\) such that \(S \cap T = \emptyset\) and the group of \((S,T)\)-units of \(L\) is \(\mathbb{Z}\)-torsion-free. The Stickelberger element and the refined regulator both depend on \(S\) and \(T\), and \(n = \# S - 1\). The conjecture predicts that \[ \theta \equiv \pm h \mathcal{R} \mod I^{n+1} \] where \(h\) is the \((S,T)\)-class-number of \(K\). The article under review proves Gross' Conjecture (in its formulation for infinite abelian extensions) when \(L\) is the maximal unramified-outside-\(S\) abelian extension of \(K = \mathbb{F}_q(x)\) in the case where \(q \not\equiv 1 \mod 4\), \(S\) consists of places of degree \(1\), and \(T\) consists of places for which the greatest common divisor of their degrees is coprime to \(q - 1\). The method uses the notions of ``tame family'' and ``universal polynomial'', which are too technical to discuss here. A key ingredient is that Stickelberger elements form a tame family and that this family is ``represented'' by a universal polynomial. In the function field case, Gross' Conjecture had already been proven in 2007; see Theorem A of \textit{D. Burns} [Invent. Math. 169, No. 3, 451--499 (2007; Zbl 1133.11063)]. (Note, however, that Burns' proof in that paper used the truth, established by Tan, of Gross' Conjecture for extensions of \(p\)-power degree where \(p\) is the residue characteristic of the given function fields.) The main benefit of the paper under review, therefore, seems to be the relative directness of the proof it provides for Gross's Conjecture (in the case in question) and the study of Stickelberger elements in terms of universal polynomials.
special values of abelian \(L\)-functions
scientific article; zbMATH DE number 5654991
scientific article