In the famous paper “Jacobi sums as ‘Grössencharaktere’,” Trans. Am. Math. Soc. 73, 487–495 (1952; Zbl 0048.27001), A. Weil provided the first family of curves over number fields (Fermat curves) whose \(L\)- functions had a meromorphic continuation to all of \(\mathbb{C}\). This was accomplished by showing that such \(L\)-functions could be factored according to the \(L\)-functions of certain generalized characters (“Grössencharakter”) of the idèle class group (or the associated “Hecke character” on the idèle group itself). Standard techniques then show that the \(L\)-functions of these Grössencharakters have meromorphic continuations and functional equations. The main part of all this is to show that Jacobi sums – which are the key to the above factorization – can be used to define such Grössencharakters. In the paper “Sommes de Jacobi et caractères de Hecke”, Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1974, 1–14 (1974; Zbl 0367.10035), A. Weil showed how to construct such characters for general abelian extensions of \(\mathbb{Q}\) but now used Gauss sums instead of Jacobi sums.
Now let \(k\) be a global function field. The Brumer-Stark conjecture is known to be true over any global function field by a theorem of Deligne [see, e.g., Chapter V of J. Tate, Les conjectures de Stark sur les fonctions \(L\) d’Artin en \(s=0\) (Prog. Math. 47) Boston etc.: Birkhäuser (1984; Zbl 0545.12009)], and the author has given a proof of this using Drinfeld modules in [“Stickelberger elements in function fields”, Compos. Math. 55, 209–239 (1985; Zbl 0569.12008)].
A main result of the beautiful and important paper under review is that the Brumer-Stark units in any finite abelian extension \(K/k\) can be aligned into a Hecke character \(\Phi\). When \(K/k\) is a cyclotomic function field (so coming directly from sign normalized rank one Drinfeld modules), \(\Phi\) is an analogue of the Gauss sum Hecke character described by Weil. The main tool used by the author is the Eisenstein reciprocity law; an analogue for function fields of the Eisenstein reciprocity law constructed by A. Weil in “La cyclotomie jadis et naguère”, Sémin. Bourbaki 1973/74, Exp. No. 452, Lect. Notes Math. 431, 318–338 (1975; Zbl 0362.12003); Enseign. Math., II. Sér. 20, 247–263 (1974; Zbl 0352.12006).
In certain instances the author’s Hecke characters can be used to define characteristic \(p\) valued \(L\)-series [see, the reviewer, “\(L\)-series of Grössencharakters of type \(A_ 0\) for function fields”, in: \(p\)-adic methods in Number Theory and Algebraic Geometry, Contemp. Math. 133, 119–139 (1992; Zbl 0941.11504) and “\(L\)-series of \(t\)-motives and Drinfeld modules”, in: The Arithmetic of Function Fields, Ohio State Univ. Math. Res. Inst. Publ. 2, 313–402 (1992; Zbl 0806.11028)]. The special values of such functions should be of substantial arithmetic interest.