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 Tata Institute of Fundamental Research 2010; 540 pp; hardcover ISBN-10: 81-8487-085-X ISBN-13: 978-81-8487-085-5 List Price: US$50 Member Price: US$40 Order Code: TIFR/16 This is the proceedings of the International Colloquium organized by the Tata Institute of Fundamental Research in January 2008, one of a series of colloquia going back to 1956. It covers a wide spectrum of mathematics, ranging over algebraic geometry, topology, automorphic forms, and number theory. Algebraic cycles form the basis for the construction of motives, and conjectures about motives depend ultimately on important problems related to algebraic cycles, such as the Hodge and the Tate conjectures. Shimura varieties provide interesting, nontrivial instances of these fundamental problems. On the other hand, the motives of Shimura varieties are of great interest in automorphic forms and number theory. This volume features refereed articles by leading experts in these fields. The articles contain original results as well as expository material. A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka. Readership Graduate students and research mathematicians interested in cycles, motives, and Shimura varieties. Table of Contents D. Arapura -- Mixed Hodge structures associated to geometric variations M. Asakura and K. Sato -- Beilinson's Tate conjecture for $$K_2$$ of elliptic surface: Survey and examples E. Ghate -- On the freeness of the integral cohomology groups of Hilbert-Blumenthal varieties as Hecke modules P. Griffiths -- Singularities of admissible normal functions G. Harder -- Arithmetic aspects of rank one Eisenstein cohomology K. Kimura -- A remark on the second Abel-Jacobi map A. Krishna and V. Srinivas -- Zero cycles on singular affine varieties M. Levine -- Tate motives and the fundamental group S.-J. Kang and J. D. Lewis -- Beilinson's Hodge conjecture for $$K_{1}$$ revisited S. Kimura and J. P. Murre -- On natural isomorphisms of finite dimensional motives and applications to the Picard motives A. Miller -- Chow motives of mixed Shimura varieties A. Neeman -- Dualizing complexes--the modern way A. Rosenschon and V. Srinivas -- The Griffiths group of the generic abelian 3-fold R. Sreekantan -- Non-Archimedean regulator maps and special values of $$L$$-functions T. Terasoma -- The Artin-Schreier DGA and the $$\mathbf{F}_p$$-fundamental group of an $$\mathbf{F}_p$$ scheme