# nLab Beilinson conjecture

### Context

#### Differential cohomology

differential cohomology

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

Michael Rapoport, Norbert Schappacher, Peter Schneider (eds.), Beilinson's Conjectures on Special Values of L-Functions Perspectives in Mathematics, Volume 4, Academic Press, Inc. 1988 (ISBN:978-0-12-581120-0)

A noncommutative analogue is considered in