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\(p\)-adic uniformization of Shimura varieties. (Uniformisation \(p\)-adique des variétés de Shimura.) (French) Zbl 0932.14013

Séminaire Bourbaki. Volume 1996/97. Exposés 820–834. Paris: Société Mathématique de France, Astérisque. 245, 307-322, Exp. No. 831 (1997).
The theory of \(p\)-adic uniformization is a theory describing certain algebraic varieties as quotients of a \(p\)-adic upper half space, or more generally, of the analytic space of a formal scheme defined over the ring of integers of a finite extension \(K\) of the \(p\)-adic numbers (or a local non Archimedean field). In this theory one attempts to replace the classical complex upper half space by a \(p\)-adic analogue: Optimally, by the \(n\) dimensional projective space over a complete algebraic closure of the \(p\)-adic numbers from which the \(K\) rational hyper-planes are removed. The paper begins with a short historical survey and describes the state of the art. The theory originated with J. Tate followed by D. Mumford [Compos. Math. 24, 129–174 (1972; Zbl 0228.14011)] who considered the case of curves over a complete local ring with reduction consisting of rational curves. It was then expanded by I. V. Cherednik [Math. USSR, Sb. 29 (1976), 55–78 (1978); translation from Mat. Sb., Nov. Ser. 100(142), 59–88 (1976; Zbl 0329.14015)] who considered the case of a Shimura curve associated to a quaternion algebra over the rational numbers that is ramified at \(p\). Its special fiber is of the form considered by Mumford. The theory was much expanded by V. G. Drinfel’d [Funct. Anal. Appl. 10, 107–115 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 29–40 (1976; Zbl 0346.14010)] and J. F. Boutot and H. Carayol [in: Courbes modulaires et courbes de Shimura, C. R. Sémin., Orsay 1987/88, Astérisque 196/197, 45–158 (1991; Zbl 0781.14010)], who interpreted the covering space as a moduli space for a certain class of \(p\)-divisible groups. Drinfel’d also pointed out to deep connections between \(p\)-adic uniformization and Langlands’ conjectures. The author explains the appearance of \(p\)-adic uniformization in the theory of Shimura curves, and the extensions found by M. Rapoport and Th. Zink [“Period spaces for \(p\)-divisible groups”, Ann. Math. Stud. 141 (1996; Zbl 0873.14039)] who showed, e.g., that the formal neighborhood of the supersingular locus in a typical moduli problem of abelian varieties is parameterized by a formal scheme. To explain this result there is a discussion of moduli problems of isogeny class of \(p\)-divisible groups with extra structure. The particular case considered by Drinfeld (loc. cit.) is discussed in detail, including the connection to the Bruhat-Tits building. A merit of this case is that one obtains an explicit description of the uniformizing formal scheme.
In the last sections of the paper the author discusses the theory of Shimura curves and the application of \(p\)-adic uniformization to it. The link to the case of moduli of \(p\)-divisible groups is that when \(p\) ramifies in the quaternion algebra all the abelian varieties with quaternionic multiplication (and suitable extra structure) over an algebraically closed field of characteristic \(p\) form a single isogeny class.
For the entire collection see [Zbl 0910.00034].

MSC:

14G35 Modular and Shimura varieties
14G22 Rigid analytic geometry
11G18 Arithmetic aspects of modular and Shimura varieties
14L05 Formal groups, \(p\)-divisible groups
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