an:03966287
Zbl 0599.14019
Carayol, Henri
Sur la mauvaise r??duction des courbes de Shimura. (Bad reduction of Shimura curves)
FR
C. R. Acad. Sci., Paris, S??r. I 296, 557-560 (1983).
00148653
1983
j
14G25 14K15 14K10 14H45
bad reduction of Shimura curves; Shimura variety; characteristic p
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.