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Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006
Shimura varieties are arithmetic quotients of locally symmetric spaces which are canonically defined over number fields. In this article, we discuss recent developments on the reciprocity law realized on cohomology groups of Shimura varieties which relate Galois representations and automorphic representations. Focus is put on the control of -adic families of Galois representations by -adic families of automorphic representations. Arithmetic geometrical ideas and methods on Shimura varieties aredoi:10.4171/022-2/16 fatcat:yudhc354gnantg5qf2ie6wltrm