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$$p$$-adic uniformization of Shimura curves: the theorems of Cherednik and Drinfeld. (Uniformisation $$p$$-adique des courbes de Shimura: Les théorèmes de Čerednik et de Drinfeld.) (French) Zbl 0781.14010
Courbes modulaires et courbes de Shimura, C. R. Sémin., Orsay/Fr. 1987-88, Astérisque 196-197, 45-158 (1991).
This is the central article of the whole volume. It gives a detailed account of the theorem of Cherednik and Drinfeld on the $$p$$-adic uniformization of Shimura curves, which in turn plays a key role in K. A. Ribet’s paper [Invent. Math. 100, No. 2, 431–476 (1990; Zbl 0773.11039)].
In the first chapter, the “nonarchimedean upper half plane” $$\Omega=\mathbb{P}^ 1(C)-\mathbb{P}^ 1 (K)$$ is introduced (here $$K$$ is a local nonarchimedean field and $$C$$ the completion of an algebraic closure of $$K)$$. The Bruhat-Tits tree $$T$$ of $$\text{PGL}_ 2(K)$$ is used to endow $$\Omega$$ with a rigid-analytic structure, and a formal scheme $$\hat\Omega$$ over the ring $${\mathcal O}$$ of integers in $$K$$ is constructed with generic fibre $$\Omega$$ and special fibre a tree of projective lines with intersection graph $$T$$. Then Drinfeld’s modular interpretation of $$\hat\Omega$$ is carefully explained, as well as the action of $$\text{PGL}_ 2(K)$$ on $$T$$, $$\Omega$$ and the functor represented by $$\hat\Omega$$.
The second chapter is devoted to a detailed proof of Drinfeld’s theorem: $$K$$ is now assumed to have characteristic 0, and a quaternion algebra $$D$$ with center $$K$$ is chosen. In this situation, Drinfeld’s theorem states that the formal scheme $$\hat\Omega \widehat{\bigotimes}_{\mathcal O} \widehat{\mathcal O}^{nr}$$ over the completion $$\widehat{\mathcal O}^{nr}$$ of the strict henselization of $${\mathcal O}$$ classifies certain “special” formal $${\mathcal O}_ D$$-modules of height 4 (or equivalently their Dieudonné modules), endowed with a certain “level structure”.
In the final chapter, the Shimura curve $$S_ U$$ associated with a compact open subgroup $$U$$ of $$\Delta^*(\mathbb{A}_ f)$$ is introduced $$(\Delta$$ is an indefinite quaternion algebra with center $$\mathbb{Q}$$, $$\mathbb{A}_ f$$ are the finite adeles over $$\mathbb{Q})$$. It is known that $$S_ U$$ can be interpreted as the moduli space of abelian surfaces with complex multiplication by $${\mathcal O}_ \Delta$$ and “level-$$U$$- structure”; in particular, $$S_ U$$ is defined over $$\mathbb{Q}$$. The complex points of $$S_ U$$ can be described as $$S_ U(\mathbb{C})=\Delta^*(\mathbb{Q}) \backslash \bigl( \mathbb{P}^ 1(\mathbb{C})-\mathbb{P}^ 1 (\mathbb{R}) \bigr) \times\Delta^*(\mathbb{A}_ f)/U$$ $$S_ U(\mathbb{C})$$ is a finite union of compact Riemann surfaces.
The theorem of Cherednik (in the formulation of Drinfeld) describes a $$p$$-adic analogue of this uniformization at a “bad” prime $$p$$ (i.e. $$p$$ divides the discriminant of $$\Delta)$$: by suitably extending the modular interpretation of $$S_ U$$ to $$\mathbb{Z} \bigl[{1\over p}\bigr]$$, a model of $$S_ U$$ over $$\mathbb{Z}_ p$$ is constructed; then by comparing this moduli functor to the one in Drinfeld’s theorem, the following description of the rigid-analytic variety $$S_ U^{p,an}$$ associated with the model of $$S_ U$$ over $$\mathbb{Z}_ p$$ is obtained: $$S_ U^{p,an}\cong \mathrm{GL}_ 2(\mathbb{Q}_ p)\backslash \bigl( \Omega \widehat{\bigotimes} \mathbb{Q}_ p ^{nr} \bigl) \times{\mathcal Z}_ U$$ with a certain adelically defined set $${\mathcal Z}_ U$$ of double cosets; it turns out that $$S_ U^{p,an}$$ is a finite union of Mumford curves.
[For the entire collection see Zbl 0745.00052.]

##### MSC:
 14G20 Local ground fields in algebraic geometry 14G35 Modular and Shimura varieties 11G09 Drinfel’d modules; higher-dimensional motives, etc. 14H25 Arithmetic ground fields for curves 14L05 Formal groups, $$p$$-divisible groups