The modern theory of the automorfas forms, flesh in which has arrived to be known like the program of Langlands, is an extraordinary unificadora force of the mathematics. It proposes fundamental relations that the arithmetical information necktie of the theory of numbers and algebraic geometry with analytical information from the harmonic analysis and representations of group. These " laws of the reciprocity, conjectured by Langlands, continue to a great extent not being tried. Nevertheless, its capacity to unite great areas of the mathematics assures that it will be a central area of study in the next years. The objective of this volume is to provide an entry point in this exciting and challenging field. It is directed, on the one hand, to the students of posgrado and the professional mathematicians who want to work in articles area. The more in particular, represents an attempt so that the reader to dominate some of the most difficult techniques. On the other hand, the book also will be useful to the mathematicians to whom it would like simply to understand something of the subject. They will be able to consult to the expositivas parts of different articles. The volume concentrates around the plan formula and Shimura varieties. These areas are in the heart of this subject, but they have been especially difficult to learn due to the lack of expositivo material. The volume tries to correct the problem. School of Summer of the Institute is based on the courses distributed in 2003 Clay de Matemáticas. Nevertheless, many of articles have been extended in the complete introduction, or for the plan of formula or the theory of the varieties of Shimura, or with some aspect of the interaction and the application of the two areas. This book is adapted for the independent study.

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