# Poincare biextension and ideles on an algebraic curve

## Abstract

The Weil pairing of two elements of the torsion of the Jacobian of an algebraic curve can be expressed in terms of the product of the local Hilbert symbols of two special ideles associated with the torsion elements of the Jacobian. On the other hand, Arbarello, De Concini, and Kac have constructed a central extension of the group of ideles on an algebraic curve in which the commutator is also equal up to a sign to the product of all the local Hilbert symbols of two ideles. The aim of the paper is to explain this similarity. It turns out that there exists a close connection between the Poincare biextension over the square of the Jacobian defining the Weil pairing and the central extension constructed by Arbarello, de Concini, and Kac. The latter is a quotient of a certain biextension associated with the central extension.

- Authors:

- V.A. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21266955

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Sbornik. Mathematics; Journal Volume: 197; Journal Issue: 1; Other Information: DOI: 10.1070/SM2006v197n01ABEH003744; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMMUTATORS; HILBERT SPACE; JACOBIAN FUNCTION; POINCARE GROUPS; TORSION

### Citation Formats

```
Gorchinskii, S O.
```*Poincare biextension and ideles on an algebraic curve*. United States: N. p., 2006.
Web. doi:10.1070/SM2006V197N01ABEH003744; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Gorchinskii, S O.
```*Poincare biextension and ideles on an algebraic curve*. United States. doi:10.1070/SM2006V197N01ABEH003744; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Gorchinskii, S O. Tue .
"Poincare biextension and ideles on an algebraic curve". United States.
doi:10.1070/SM2006V197N01ABEH003744; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21266955,
```

title = {Poincare biextension and ideles on an algebraic curve},

author = {Gorchinskii, S O},

abstractNote = {The Weil pairing of two elements of the torsion of the Jacobian of an algebraic curve can be expressed in terms of the product of the local Hilbert symbols of two special ideles associated with the torsion elements of the Jacobian. On the other hand, Arbarello, De Concini, and Kac have constructed a central extension of the group of ideles on an algebraic curve in which the commutator is also equal up to a sign to the product of all the local Hilbert symbols of two ideles. The aim of the paper is to explain this similarity. It turns out that there exists a close connection between the Poincare biextension over the square of the Jacobian defining the Weil pairing and the central extension constructed by Arbarello, de Concini, and Kac. The latter is a quotient of a certain biextension associated with the central extension.},

doi = {10.1070/SM2006V197N01ABEH003744; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

number = 1,

volume = 197,

place = {United States},

year = {Tue Feb 28 00:00:00 EST 2006},

month = {Tue Feb 28 00:00:00 EST 2006}

}