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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