This volume covers the proceedings of the International Colloquium organised 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, like the Hodge and the Tata 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 book contains refereed articles by leading experts in these fields, containing original results, as well as expository material, on these areas.
Mixed Hodge Structures Associated to Geometric Variations / Beilinsons Tata Conjecture for K2 of Elliptic Surface: Survey and Examples / On the Freeness of the Integral Cohomology Groups of Hilbert-Blumenthal Varieties as Hecke Modules / Singularities of Admissable Normal Functions / Arithmetic Aspects of Rank one Eisenstein Cohomology / A Remark on the Second Abel-Jacobi Map / Zero Cycles on Singular Affine Varieties / Tata Motives and the Fundamental Group / Beilinsons Hodge Conjecture for K1 Revisited / On Natural Isomorphisms of Finite Dimensional Motives and Applications to the Picard Motives / Chow Motives of Mixed Shimura Varieties / Dualizing Complexes: The Modern Way / The Griffiths Group of the Generic Abelian 3-fold / Non-Archimedean Regulator Maps and Special Values of L-functions / The Artin-Schreier DGA and the Fp-fundamental group of an Fp.