Spencer Bloch

Algebraic cycles and higher K-theory

Here G,(X) is the Grothendieck group of coherent sheaves on X [13], gr; refers to the graded group defined by the y-filtration on G,(X) (cf. Kratzer [14], Soult [20]), and CH’(X) is the Chow group of codimension i algebraic cycles defined by Fulton [9]. The left-hand isomorphism is a formal consequence of the existence of a I-structure on G,(X) while the existence of r is the central theme of the B-F-M RR theorem. The main purpose of this paper is to define a theory of higher Chow groups CH*(X, n), n 2 0, so as to obtain isomorphisms

L-Functions and Tamagawa Numbers of Motives

The notion of a motif was first defined and studied by A. Grothendieck, and this paper is an attempt to understand some of the implications of his ideas for arithmetic. We will formulate a conjecture on the values at integer points of L-functions associated to motives. Conjectures due to Deligne and Beilinson express these values “modulo Q* multiples” in terms of archimedean period or regulator integrals. Our aim is to remove the Q* ambiguity by defining what are in fact Tamagawa numbers for motives. The essential technical tool for this is the Fontaine-Messing theory of p-adic cohomology. As evidence for our Tamagawa number conjecture, we show that it is compatible with isogeny, and we include strong results due to one of us (Kato) for the Riemann zeta function and for elliptic curves with complex multiplication.

p-adic Etale Cohomology

What follows is a report on joint work with O. Gabber and K. Kato.* A manuscript with complete proofs exists and is currently being revised. For compelling physical reasons (viz. time, space, and distance) however, I will give here only statements of results; and my coauthors have not had the opportunity to correct any stupidities which may have slipped in. The conjectures in §3 are my own. I like to think that this research has been strongly influenced by the work of Shafarevich, both by his work on algebraic geometry in characteristic p and by his work on arithmetical algebraic geometry. In fact, recently Ogus has used these results to apply the basic Rudakov-Shafarevich result on existence and smoothness of moduli for K3 surfaces in characteristic p to the study of the moduli space when p = 2.

On Motives associated to graph polynomials

The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions.

Higher regulators, algebraic K-theory, and zeta functions of elliptic curves

Introduction Tamagawa numbers Tamagawa numbers. Continued Continuous cohomology A theorem of Borel and its reformulation The regulator map. I The dilogarithm function The regulator map. II The regulator map and elliptic curves. I The regulator map and elliptic curves. II Elements in

Algebraic

K_2(E)

K

of an elliptic curve

-theory and crystalline cohomology

E

The elliptic dilogarithm for the sunset graph

A regulator formula Bibliography Index

A Feynman integral via higher normal functions

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