# nLab Beilinson conjecture

### Context

#### Differential cohomology

differential cohomology

## Ingredients

• cohomology

• differential geometry

• ## Connections on bundles

• connection on a bundle

• curvature

• Chern-Weil theory

• ## Higher abelian differential cohomology

• differential function complex

• differential orientation

• ordinary differential cohomology

• differential K-theory

• differential elliptic cohomology

• differential cobordism cohomology

• ## Higher nonabelian differential cohomology

• Chern-Weil theory in Smooth∞Grpd

• ∞-Chern-Simons theory

• ## Fiber integration

• higher holonomy

• fiber integration in differential cohomology

• ## Application to gauge theory

• gauge theory

• gauge field

• quantum anomaly

• #### Complex geometry

complex geometry

# Contents

## Idea

Beilinson’s conjectures (Beilinson 85) conjecture for arithmetic varieties over number fields

1. that the realification of the Beilinson regulator exhibits an isomorphism between the relevant algebraic K-theory/motivic cohomology groups and Deligne cohomology (ordinary differential cohomology) groups;

2. induced by this that special values of the (Hasse-Weil-type) L-function are proportional to the Beilinson regulator, in analogy with the class number formula and the Birch and Swinnerton-Dyer conjecture

The Beilinson conjecture for special values of L-functions follows the Birch and Swinnerton-Dyer conjecture and Pierre Deligne‘s conjecture on special value of L-functions.

## References

The original articles are

Reviews include

A noncommutative analogue is considered in

Last revised on October 24, 2014 at 20:16:07. See the history of this page for a list of all contributions to it.