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Fonction zêta locale et fonction L de formes automorphes. (French) Zbl 0559.14012
Variétés de Shimura et fonctions L, Publ. Math. Univ. Paris VII, No. 6, 83-130 (1979).
[For the entire collection see Zbl 0517.00003.]
In another article in the same volume, 11-20 (1979; Zbl 0551.14008) containing a ”modern” version of the Kronecker-Weber congruence formulae, the author discussed the Eichler-Shimura congruence generalizing Weber’s classical result on the reduction modulo p of the modular equation and an application to the calculation of the local zeta functions of modular varieties for ”good” primes. The present paper seeks to give an expression for the local zeta function at p of a Shimura variety \({}_ KS\) with ”level structure K”, associated with a quaternion algebra B, over a totally real number field E, split at infinity and at a (rational) prime p (unramified in E), in terms of local L-functions attached to automorphic forms connected with the group \(G=Res_{E/{\mathbb{Q}}}(B^{opp})^{\times}\) (and its centre Z). The first three sections are concerned with the (essentially combinatorial) question of calculating the number of fixed points of the powers of the Frobenius operator on the \({\bar {\mathbb{F}}}_ p\)-rational points of \({}_ KS\), while, in the next section, this is related, via the ”trace formula”, to the trace of a natural representation of the adelic group G(\({\mathbb{A}})\). The last section contains a reinterpretation, to express the local zeta function of \({}_ KS\) at p in terms of the corresponding local L-factors attached to automorphic forms.
Reviewer: S.Raghavan

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11S40 Zeta functions and \(L\)-functions
11F70 Representation-theoretic methods; automorphic representations over local and global fields
14G25 Global ground fields in algebraic geometry
14G20 Local ground fields in algebraic geometry
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties