nLab Beilinson's Conjectures on Special Values of L-Functions


This entry is about the book

on Deligne cohomology and Beilinson's conjectures on special values of L-functions.

(The book is out of print and unavailable from the publisher, even electronically.)



Every Spring and Fall, in the woods of the Black Forest, mathematicians from West-Germany and other countries gather at Oberwolfach for the “Arbeitsgemeinschaft Geyer-Harder” to teach themselves in a joint effort new theories or results. At each meeting the topic and organizers of the following Arbeitsgemeinschaft are chosen after an evening of discussions by democratic vote. The organizers then prepare a detailed program for the conference which is posted in the mathematical institutes. Everyone interested in taking part in the meeting has to volunteer for one of the talks in the program. The organizers then choose the actual speakers. The Arbeitsgemeinschaft of April 1986 was devoted to the work of Beilinson and Bloch on regulators. The meeting centered around Beilinson’s article “Higher regulators and values of L-functions”. It was generally felt at the conference that the joint effort made to understand this fundamental paper should not be lost, and should be helpful to others. Hence this volume. Most of its chapters are based on talks at the meeting; others have been added in order to give a coherent account of the conjectures and some of the known evidence for them. In the name of all participants we want to express our gratitude to the Mathematisches Forschungsinstitut Oberwolfach, which makes the Arbeitsgemeinschaft possible.


Michael Rapoport

Norbert Schappacher

Peter Schneider

Program proposal

Introduction to the Beilinson Conjectures

Peter Schneider.

§1 Complex L-functions

§2 Deligne cohomology

§3 Absolute cohomology

§4 Chern classes

§5 The conjectures

§6 Further hints

Deligne’s Conjecture

Maria Heep?, Uwe Weselmann?.

§1 Motives

§2 Duality, functional equation, criticial values

§3 Deligne’s periods

§4 Beilinson’s periods

Deligne-Beilinson Cohomology

Hélène Esnault, Eckart Viehweg.

§1 The Deligne cohomology

§2 The Deligne-Beilinson complex

§3 Products

§4 Relative cohomology

§5 Extensions and complements

§6 The cycle map in the de Rham cohomology

§7 The cycle map in the Deligne cohomology

§8 Chern classes in the Deligne-Beilinson cohomology

λ\lambda-Rings and Adams Operations in Algebraic K-Theory.

Wolfgang K. Seiler?.

The Theorem of Riemann-Roch

Günter Tamme.

§1 The λ-ring structure and the Chern character for K-theory with supports

§2 Riemann-Roch without denominators

§3 The smooth Riemann-Roch theorem for K-theory with supports

§4 The singular Riemann-Roch

§5 Absolute cohomology and homology

Comparison of the Regulators of Beilinson and of Borel.

Michael Rapoport.

The Beilinson Conjecture for Algebraic Number Fields.

Jürgen Neukirch.

Part I: Regulators and values of Artin L-series

§1 Regulators for algebraic number fields

§2 Regulators for Artin motives

§3 L-series of Artin motives

§4 Dirichlet L-series

§5 Regulators of Dirichlet motives

Part II: The regulator map for cyclotomic fields

§1 The main theorem

§2 Universal symbols

§3 Special symbols

§4 Reduction to the main lemma

§5 Proof of the main lemma for n = 1

§6 Proof of the main lemma for n ≥ 2

On the Beilinson Conjectures for Elliptic Curves with Complex Multiplication

Christopher Deninger, Kay Wingberg?.

§1 A formula for the regulator of curves

§2 A weakened version of the Beilinson conjecture for elliptic curves

§3 Calculating ω,[α,β] calD\langle\omega,[\alpha,\beta]_{\cal D}\rangle for elliptic curves over bfR\bf R

§4 Relations between the L-function of an elliptic curve over Q with complex multiplication and Eisenstein-Kronecker-Lerch series

§5 The absolute cohomology

Beilinson’s Theorem on Modular Curves.

Norbert Schappacher, Anthony J. Scholl?.

§1 The Theorem

§2 Transformation of L-values

§3 Eisenstein series and modular units

§4 Whittaker functions and L-factors

§5 Evaluation of the regulator integral

§6 Non-vanishing of the regulator integral

§7 Integrality

Deligne Homology, Hodge-D-Conjecture, and Motives

Uwe Jannsen.

Last revised on March 15, 2021 at 09:44:39. See the history of this page for a list of all contributions to it.