The points on a Shimura variety modulo a prime of good reduction. (English) Zbl 0821.14016

Milne, James S.

The zeta functions of Picard modular surfaces. CRM Workshop, Montreal / Can. 1988, 151-253 (1992).

[For the entire collection see Zbl 0752.00024.]
Let \(\text{Sh} (G,X)\) be the Shimura variety associated to a reductive group \(G\) over \(\mathbb{Q}\), and let \(\text{Sh}_ p (G,X) = \text{Sh} (G,X)/K_ p\), where \(K_ p\) is a compact open subgroup of \(G(\mathbb{Q}_ p)\). Let \(E\) be a reflex field of \(\text{Sh} (G,X)\), and let \(v\) be a prime of \(E\) lying over \(p\). If \(\mathbb{Q}^{\text{al}}_ p\) is the algebraic closure of \(E_ v\) and \(\mathbb{Q}^{\text{un}}_ p\) is the maximal unramified extension of \(\mathbb{Q}_ p\) contained in \(\mathbb{Q}^{\text{al}}_ p\), then \(\mathbb{F}\) denotes the residue field of \(\mathbb{Q}^{\text{un}}_ p \subset \mathbb{Q}^{\text{al}}_ p\). Let \({\mathfrak P}\) be the pseudomotivic groupoid associated with the Tannakian category of motives over \(\mathbb{F}\), and let \({\mathfrak G}_ G\) be the neutral groupoid defined by \(G\). Then a homomorphism \(\varphi : {\mathfrak G}_ G \to {\mathfrak P}\) defines a triple \((S (\varphi), \Phi (\varphi), \times (\varphi))\), where \(S(\varphi)\) is a set of the form \(I_ \varphi (\mathbb{Q})^ - \backslash X^ p (\varphi) \times X_ p (\varphi)\), \(\Phi (\varphi)\) is a Frobenius operator, and \(X(\varphi)\) is an action of \(G(\mathbb{A}^ p_ f)\) on \(S(\varphi)\) commuting with the action of \(\Phi (\varphi)\). Then the conjecture of R. P. Langlands and M. Rapoport [J. Reine Angew. Math. 378, 113-220 (1987; Zbl 0615.14014)] can be stated as \[ \bigl( \text{Sh}_ p (\mathbb{F}), \Phi, \times \bigr) \approx \coprod_ \varphi \bigl( S (\varphi), \Phi (\varphi), \times (\varphi) \bigr), \] where the disjoint union is over a certain set of isomorphism classes of \(\varphi\). Let \({\mathcal V} (\xi)\) be the local system on \(\text{Sh} (X)\) defined by a representation \(\xi : G \to GL (V)\) of \(G\). Let \({\mathcal T} (g)\) be the Hecke operator defined by \(g \in G (\mathbb{A}_ f^ p)\), and let \({\mathcal T} (g)^{(r)}\) be the composite of \({\mathcal T} (g)\) with the \(r\)-th power of the Frobenius correspondence. In this paper the author derives from the conjecture of Langlands and Rapoport the formula for the sum \[ \sum_{t'} \text{Tr} \bigl( {\mathcal T} (g)^{(r)} | {\mathcal V}_ t (g) \bigr), \] where \(t'\) runs over the elements of \(\text{Sh}_{K \cap gKg^{-1}} (G,X) (\mathbb{F})\) such that \({\mathcal T} (g)(t') = t\), as a sum of products of certain orbital integrals. He also introduces the notion of an integral canonical model for a Shimura variety, extends the conjecture of Langlands and Rapoport to Shimura varieties defined by groups whose derived group is not simply connected, and reviews results of R. Kottwitz concerning the stabilization.

Reviewer: Min Ho Lee (Cedar Falls)

MSC:

14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G35	Modular and Shimura varieties
11F70	Representation-theoretic methods; automorphic representations over local and global fields
14G15	Finite ground fields in algebraic geometry

Keywords:

zeta functions; Picard modular surfaces; \(L\)-functions; automorphic representations; Shimura variety; Tannakian category of motives; Frobenius operator; conjecture of Langlands and Rapoport

Citations:

Zbl 0752.00024; Zbl 0615.14014