Motives, Polylogarithms and Hodge Theory (Part I: Motives and Polyogarithms)
Edited by: Fedor Bogomolov & Ludmil Katzarkov
ISBN:1-57146-090-X
Year
Published: 2002
Page: 414
Binding:
Hardcover
Price: $65
(or $100 for a set consisting of both Part 1 and 2)
Description
The present
volume contains papers of the participants in the International Press
Conference
on Motives, polylogarithms and non-abelian Hodge theory
which took
place at UC Irvine in June 1998. The conference commemorated
the
twentieth anniversary of the remarkable
"Higher regulators, algebraic K-theory and zeta functions of
elliptic curves". The conference presented some of the best recent research in algebraic
K-theory, Hodge theory, motivic cohomology
and polylogarithms. The research program of the
conference was organized around three main lecture series:
VladimirVoevodsky taught a minicourse overviewing
the recent developments in motivic cohomology and motivic homotopy theory; Don Zagier
lectured on new results describing the periods of holomorphic
and non-holomorphic modular forms; and Carlos Simpson
lectured on the theory of geometric n-stacks and its applications to the variational aspects of non-abelian
Hodge theory.
Table of Contents
I. MOTIIVES
Open Problems in the Motivic Stable Homotopy Theory, I,
By Vladimir Voevodsky
1 Introduction
2 Slice filtration
3 Main conjectures
4 Slice-wise cellular spectra
5 Reformulations in terms of rigid homotopy groups
6 Rigid homology and rigid
7 Slice spectral sequence and
convergence problems
8 Possible strategies of the proof
Remarks on n-motives and correspondences at the generic point,
By AIexander
Beilinson
1 Introduction
2 The setting
3 Some natural injective n-motives
4 Some applications
Relative algebraic differential characters,
By Spencer Bloch and Helene Esnault
1 Introduction
2 Relative Cohomology
3 Splitting Principle
4 Universal construction
via the Weil algebra
5 The image of C2 in a family of
curves
Commuting elements in Galois groups of function fields,
By Fedor Bogomolov and Yuri Tschinkel
2 Classes of functions
3 Reductions
4 AF-functions and geometry
5 Galois theory
6 Valuations
Mixed Hodge Structures and Iterated Integrals, I,
by Zdzislaw Wojtkowiak
0 Introduction
1 Monodromy
of iterated integrals
2 Mixed Hodge structures
3 Cosimplicial
objects and mixed motives
4 Coefficients
5 The conjecture (real form)
6 Proof of the conjecture
7 The conjecture (complex form)
8 Functional equations on P1(C) \ {O,1,oo}
9 Coefficients on P1Q\ {O,1,oo}
10 Examples
11 Iterated extensions
12 Exotic zeta functions
13 Problems
Appendix A. Tangential base points
and monodromy of iterated integrals
Appendix B. Group Zariski
closure
II POLYLOGARITHMS AND SPECIAL
FUNCTIONS
Traces of singular moduli,
by Don Zagier
Introduction
1 The trace of j(α)
2 A recursion for the numbers t(d)
3 Proof of the first recursion for t(d)
4 Proof of the second recursion for t(d)
5 Relation to Borcherds's
theorem
6 Hecke
operators
7 First generalization: other discriminants
8 Second generalization: other groups
9 Third generalization: other weights
Explicit Regulator maps on polylogarithmic motivic complexes,
By A. B. Goncharov
1 Introduction
2 The main result
3 Arakelov motivic complexes: examples
4 Proofs
An explicit formula for the motivic elliptic polylogarithm,
By Andrey
Levin
1 Introduction
2 The basic functions and symbols
3 Explicit expressions for functions
Ф for an elliptic curve over a field
On the Eisenstein symbol,
By Jorg Wildeshaus
0 Introduction
1 The elliptic motivic
polylogarithm
2 The formalism of elliptic Bloch
groups
3 The proofs