The Points on a Shimura Variety modulo a Prime of Good Reduction
نویسندگان
چکیده
We explain, in the case of good reduction, the conjecture of Langlands and Rapoport describing the structure of the points on the reduction of a Shimura variety (Langlands and Rapoport 1987, 5.e, p169), and we derive from it the formula conjectured in (Kottwitz 1990, 3.1), which expresses a certain trace as a sum of products of (twisted) orbital integrals. Also we introduce the notion of an integral canonical model for a Shimura variety, and we extend the conjecture of Langlands and Rapoport to Shimura varieties defined by groups whose derived group is not simply connected. Finally, we briefly review Kottwitz’s stabilization of his formula.
منابع مشابه
The basic stratum of some simple Shimura varieties
Under simplifying hypotheses we prove a relation between the l-adic cohomology of the basic stratum of a Shimura variety of PEL-type modulo a prime of good reduction of the reflex field and the cohomology of the complex Shimura variety. In particular we obtain explicit formulas for the number of points in the basic stratum over finite fields. We obtain our results using the trace formula and tr...
متن کاملThe Conjecture of Langlands and Rapoport for Certain Shimura Varieties of Non-rational Weight
The conjecture of Langlands and Rapoport provides a precise description of the points on a Shimura variety modulo a prime of good reduction. We reene the conjecture so that it also describes the points on the reduction of each connected component of the Shimura variety, and we prove (under some hypotheses) that the validity of the reened conjecture for a Shimura variety depends only on the asso...
متن کاملFinal Version; Final Form. Shimura Varieties and Motives
Deligne has expressed the hope that a Shimura variety whose weight is defined over Q is the moduli variety for a family of motives. Here we prove that this is the case for “most” Shimura varieties. As a consequence, for these Shimura varieties, we obtain an explicit interpretation of the canonical model and a modular description of its points in any field containing the reflex field. Moreover, ...
متن کاملOn the Conjecture of Langlands and Rapoport
FORENOTE (2007): The remarkable conjecture of Langlands and Rapoport (1987) gives a purely group-theoretic description of the points on a Shimura variety modulo a prime of good reduction. In an article in the proceedings of the 1991 Motives conference (Milne 1994, §4), I gave a heuristic derivation of the conjecture assuming a sufficiently good theory of motives in mixed characteristic. I wrote...
متن کاملAn Algorithm for Modular Elliptic Curves over Real Quadratic Fields
Let F be a real quadratic field with narrow class number one, and f a Hilbert newform of weight 2 and level n with rational Fourier coefficients, where n is an integral ideal of F . By the Eichler-Shimura construction, which is still a conjecture in many cases when [F : Q] > 1, there exists an elliptic curve Ef over F attached to f . In this paper, we develop an algorithm that computes the (can...
متن کامل