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Sur la mauvaise réduction des courbes de Shimura. (Bad reduction of Shimura curves). (French) Zbl 0599.14019
Let B/F be a quaternion algebra over a totally real number field which splits at exactly one infinite prime. Let \(K\subset B^{\times}({\mathbb{A}}_ f)\) be an open and compact subgroup. Denote by \(M_ K\) the Shimura variety associated with B and K. Fix a prime \({\mathfrak p}\) of F where B splits. The reduction of \(M_ K\) at \({\mathfrak p}\) has been studied by several authors if K is maximal compact at \({\mathfrak p}\). In the contrary case nothing was known except for \(B=M_ 2({\mathbb{Q}})\), because the usual interpretation of \(M_ K\) as a moduli scheme fails in characteristic p. Following an idea of Drinfel’d the author gives a suitable interpretation as a moduli scheme and obtains a description of the reduction.

14G25 Global ground fields in algebraic geometry
14K15 Arithmetic ground fields for abelian varieties
14K10 Algebraic moduli of abelian varieties, classification
14H45 Special algebraic curves and curves of low genus