Zeta functions of Shimura varieties. (English) Zbl 0827.11033

Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 2, 525-571 (1994).
The present paper is a survey article on the conjectured relation originated by the work of Eichler-Shimura and Langlands between zeta- functions of Shimura varieties and automorphic \(L\)-functions, and its background. The authors start with discussing \(L\)-groups, functoriality, Shimura varieties, automorphic representations, multiplicity formulas, the generalized Ramanujan-Petersson conjecture and (what is called) \(L\)- resp. \(A\)-parameters. Then they give the general zeta function conjecture which gives an expression of the \(L\)-function of a compatible system of \(\lambda\)-adic Galois representations attached to an isotypical component with respect to the action of the Hecke algebra of the intersection cohomology of a compactified Shimura variety in terms of automorphic \(L\)- functions. In the final section they discuss various examples: modular curves, the groups \(\text{GL}_2\), \(\text{GSp}_4\), \(U_n\) and Picard modular surfaces.
As the authors point out, they have not included a discussion of endoscopic groups or the stable trace formula, since the latter is not absolutely necessary to understand the main conjecture. In the context of the group \(\text{GSp}_4\), we would also like to mention the recent article of R. Weissauer, “The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula)” (preprint), in which the generalized Ramanujan-Petersson conjecture for holomorphic Siegel cusp forms of genus 2 and weight \(\geq 3\) which are not CAP is proved.
For the entire collection see [Zbl 0788.00054].


11F70 Representation-theoretic methods; automorphic representations over local and global fields
14G35 Modular and Shimura varieties
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G18 Arithmetic aspects of modular and Shimura varieties
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F25 Hecke-Petersson operators, differential operators (one variable)