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.

Table of Contents

• S. Kleiman -- The standard conjectures
• N. M. Katz -- Review of $\ell$-adic cohomology
• J. H. M. Steenbrink -- A summary of mixed Hodge theory
• L. Illusie -- Crystalline cohomology
• J. Tate -- Conjectures on algebraic cycles in $\ell$-adic cohomology
• M. Saito -- Some remarks on the Hodge type conjecture
• N. M. Katz -- Independence of $\ell$ and weak Lefschetz
• P. Deligne -- Décompositions dans la catégorie dérivée
• H. Gillet and C. Soulé -- Arithmetic analogs of the standard conjectures
• Chow groups, $K$-theory and motivic cohomology
• P. Deligne -- A quoi servent les motifs?
• A. J. Scholl -- Classical motives
• K. Künnemann -- On the Chow motive of an abelian scheme
• D. R. Grayson -- Weight filtrations in algebraic $K$-theory
• S. Bloch -- An elementary presentation for $K$-groups and motivic cohomology
• U. Jannsen -- Motivic sheaves and filtrations on Chow groups
• S. Lichtenbaum -- Motivic complexes
• M. Saito -- On the bijectivity of some cycle maps
• Motivic Galois groups
• L. Breen -- Tannakian categories
• J.-P. Serre -- Propriétés conjecturales des groupes de Galois motiviques et des représentations $\ell$-adiques
• J. S. Milne -- Motives over finite fields
• A. A. Panchishkin -- Motives for absolute Hodge cycles
• N. Schappacher -- CM motives and the Taniyama group
• P. Deligne -- Structures de Hodge mixtes réelles
• $L$-functions
• C. Deninger -- $L$-functions of mixed motives
• B. H. Gross -- $L$-functions at the central critical point
• J. Nekovář -- Beilinson's conjectures
• A. J. Scholl -- Height pairings and special values of $L$-functions
• J.-M. Fontaine and B. Perrin-Riou -- Autours des conjectures de Bloch et Kato: Cohomologie galoisienne et valeurs de fonctions $L$
• C. Deninger -- Motivic $L$-functions and regularized determinants
• M. Schröter and C. Soulé -- On a result of Deninger concerning Riemann's zeta function