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Introduction to Shimura Varieties - James Milne

Introduction to Shimura MilneOctober 23, 2004; revised September 16, 2017 AbstractThis is an Introduction to the theory of Shimura Varieties , or, in other words, to thearithmetic theory of automorphic functions and holomorphic automorphic forms. Inthis revised version, the numbering is unchanged from the original published versionexcept for symmetric domains ..52 Hodge structures and their classifying spaces ..223 Locally symmetric Varieties ..324 Connected Shimura Varieties ..425 Shimura Varieties ..526 The Siegel modular variety ..677 Shimura Varieties of Hodge type ..768 PEL Shimura Varieties ..799 General Shimura Varieties ..9010 Complex multiplication: the Shimura Taniyama formula ..9611 Complex multiplication: the main theorem .. 10612 Definition of canonical models.

The arithmetic properties of elliptic modular functions and forms were extensively studied in the 1800s, culminating in the beautiful Kronecker Jugendtraum. Hilbert emphasized the importance of extending this theory to functions of several variables in the twelfth of his famous problems at the International Congress in 1900.

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