## Values of abelian $$L$$-functions at negative integers over totally real fields.(English)Zbl 0434.12009

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 P. Cassou-Nogues [Invent. Math. 51, 29–59 (1979; Zbl 0408.12015)] and 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 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 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.
Reviewer: Kenneth A. Ribet

### MSC:

 11S40 Zeta functions and $$L$$-functions 11R42 Zeta functions and $$L$$-functions of number fields 11R80 Totally real fields 11F80 Galois representations 11F33 Congruences for modular and $$p$$-adic modular forms 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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