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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.

MSC:
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
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