Titlebook: Hodge Cycles, Motives, and Shimura Varieties; Pierre Deligne,James S. Milne,Kuang-yen Shih Book 1982 Springer-Verlag Berlin Heidelberg 198

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General Introduction,Let X be a smooth projective variety over ℂ. Hodge conjectured that certain cohomology classes on X are algebraic. The work of Deligne that is described in the first article of this volume shows that, when X is an abelian variety, the classes considered by Hodge have many of the properties of algebraic classes.

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Notations and Conventions,The ring of finite adèles, .⊗ φ, of φ is denoted by IA., and IA denotes the full ring of adèles IR. × IA.. For E a number field, IA. and IA. denote E ⊗. IA. and E ⊗ IA. The group of idèles of E is IA., and the idèle class group is C. =IA./E..

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Hodge Cycles and Crystalline Cohomology,This paper is a collection of musings about several questions related to crystalline cohomology that have plagued me for the past few years. It contains many more conjectures than proofs, and my justification for publishing is the hope that others will find the problems as intriguing as I did but perhaps have more success in solving them.

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Tannakian Categories,iliar operations on the category .. of finite-dimensional vector spaces over a field k. What complicates this is the necessity of including enough constraints so that, whenever an obvious isomorphism (e.g., . exists in .., a unique isomorphism is constrained to exist also in the abstract setting.