\(p\)-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.]

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.]

Reviewer: Frank Herrlich (Karlsruhe)

##### 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 |