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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 £-functions associated to motives. Conjectures due to Deligne and Beilinson express these values "modulo Q* multiples" in… (More)

- Spencer J. Bloch
- 1994

We show that, assuming a rather innocuous looking ”moving lemma” called Theorem A, this complex is an exact couple, and the resulting spectral sequence has the form (0.1.1). §2 §6 of the paper are devoted to proving theorem A. Finally, in §7, we prove that t≥1(Z (Spec(F ), )[−4]), the higher Chow complex shifted to the right 4 steps and then truncated so… (More)

- Spencer J. Bloch
- 2006

A notion of additive dilogarithm for a field k is introduced, based on the K-theory and higher Chow groups of the affine line relative to 2(0). Analogues of the K2-regulator, the polylogarithm Lie algebra, and the `-adic realization of the dilogarithm motive are discussed. The higher Chow groups of 0-cycles in this theory are identified with the Kähler… (More)

A report on recent results and outstanding problems concerning motives associated to graphs.

- Spencer J. Bloch, HÉLÈNE ESNAULT
- 1996

0.1. Secondary (Chern-Simons) characteristic classes associated to bundles with connection play an important role in differential geometry. We propose to investigate a related construction for algebraic bundles. Non-flat algebraic connections for bundles on complex projective manifolds are virtually non-existent (we know of none), and a deep theorem of… (More)

- Hélène ESNAULT, Spencer J. Bloch, Hélène Esnault
- 2004

Homology with values in a connection with possibly irregular singular points on an algebraic curve is defined, generalizing homology with values in the underlying local system for a connection with regular singular points. Integration defines a perfect pairing between de Rham cohomology with values in the connection and homology with values in the dual… (More)

- Spencer J. Bloch, HÉLÈNE ESNAULT
- 2005

Local Fourier transforms, analogous to the l-adic local Fourier transforms [14], are constructed for connections over k((t)). Following a program of Katz [12], a meromorphic connection on a curve is shown to be rigid, i.e. determined by local data at the singularities, if and only if a certain infinitesimal rigidity condition is satisfied. As in [12], the… (More)

— We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral; one based on an interpretation of the integral as an inhomogeneous solution of a… (More)

— We study the sunset graph defined as the scalar two-point self-energy at two-loop order. We evaluate the sunset integral for all identical internal masses in two dimensions. We give two calculations for the sunset amplitude; one based on an interpretation of the amplitude as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and… (More)

- Spencer J. Bloch
- 2005

Let G be an algebraic group defined over a number field k. By choosing a lifting of G to a group scheme over 6' s c k, the ring of S-integers for some finite set of places S of k, we may define G(C,~), where (5~, c k~ is the ring of integers in the vadic completion of k for all non-archimedean places vr In this way, we can define the adelic points G(Ak).… (More)