Motives, Part 1

American Mathematical Soc., 1994 - Mathematics - 747 pages
Motives were introduced in the mid-1960s by Grothendieck to explain the analogies among the various cohomology theories for algebraic varieties, to play the role of the missing rational cohomology, and to provide a blueprint for proving Weil's conjectures about the zeta function of a variety over a finite field. Over the last ten years or so, researchers in various areas--Hodge theory, algebraic $K$-theory, polylogarithms, automorphic forms, $L$-functions, $\ell$-adic representations, trigonometric sums, and algebraic cycles--have discovered that an enlarged (and in part conjectural) theory of mixed'' motives indicates and explains phenomena appearing in each area. Thus the theory holds the potential of enriching and unifying these areas. This is one of two volumes containing the revised texts of nearly all the lectures presented at the AMS-IMS-SIAM Joint Summer Research Conference on Motives, held in Seattle, in 1991. A number of related works are also included, making for a total of forty-seven papers, from general introductions to specialized surveys to research papers.

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Contents

 PART 3 Review of éadic Cohomology 21 A Summary of Mixed Hodge Theory 31 Crystalline Cohomology 43 Conjectures on Algebraic Cycles in ladic Cohomology 71 Some Remarks on the Hodge Type Conjecture 85 Independence of Z and Weak Lefschetz 101 Décompositions dans la catégorie dérivée 115
 Motivic Complexes 301 On the Bijectivity of Some Cycle Maps 315 Tannakian Categories 337 Propriétés conjecturales des groupes de Galois motiviques et 377 Motives over Finite Fields 401 Motives for Absolute Hodge Cycles 461 Motives and the Taniyama Group 482 Structures de Hodge mixtes réelles 509

 Arithmetic Analogs of the Standard Conjectures 129 A quoi servent les motifs? 143 Classical Motives 163 cohomologie galoisienne 186 On the Chow Motive of an Abelian Scheme 189 Weight Filtrations in Algebraic K Theory 207 An Elementary Presentation for K Groups and Motivic Cohomology 239 Motivic Sheaves and Filtrations on Chow Groups 245
 LFunctions of Mixed Motives 517 LFunctions at the Central Critical Point 527 Beilinsons Conjectures 537 Height Pairings and Special Values of LFunctions 570 Report on mod Representations of GalQQ 639 Motivic LFunctions and Regularized Determinants 707 On a Result of Deninger Concerning Riemanns Zeta Function 745 Copyright