Values of abelian \(L\)-functions at negative integers over totally real fields
The authors develop the theory of \(p\)-adic Hilbert modular forms and apply this theory to the study of abelian \(L\)-values over totally real fields. The paper begins with a long, detailed introduction, which summarizes the history and contents of the paper and states the main theorems concerning modular forms and congruences among \(L\)-values. Concerning the latter, the reader is referred to recent papers of \textit{P. Cassou-Nogues} [Invent. Math. 51, 29--59 (1979; Zbl 0408.12015)] and \textit{D. Barsky} [Groupe Étude Anal. Ultramétrique, 5e Année 1977/78, Exp. No. 16, 23 p. (1978; Zbl 0406.12008)] for an alternate method based on formulas of \textit{T. Shintani} [J. Fac. Sci., Univ. Tokyo, Sect. I A 23, 393--417 (1976; Zbl 0349.12007)]. (This method is the subject of a paper by \textit{N. Katz} which is about to appear in [Math. Ann. 255, 33--43 (1981; Zbl 0497.14006)]) An article of the second author [Journées arithmétiques de Luminy, 1978, Astérisque 61, 177--192 (1979; Zbl 0408.12016)] compares the congruences obtained by the two different methods and explains how these congruences may be used to construct \(p\)-adic \(L\)-functions.
scientific article; zbMATH DE number 3676017
scientific article