The Weil pairing and the Hilbert symbol
type: Item
instance of: scholarly article
title: The Weil pairing and the Hilbert symbol
zbMATH Open document ID: 0854.11031
DOI: 10.1007/BF01444229
author: Everett W. Howe
published in: Mathematische Annalen
publication date: 1997-01-19T00:00:00Z
full work available at URL: https://eudml.org/doc/182995
review text: Let \(C\) be a curve over a field \(k\), and suppose \(D\) and \(E\) are degree-zero divisors on \(C\) that represent \(m\)-torsion points on the Jacobian \(J\) of \(C\), so that \(mD= \text{div } f\) and \(mE=\text{div } g\) for some functions \(f\) and \(g\) on \(C\). The Weil pairing \(e_m\) on the \(m\)-torsion of \(J\), applied to the torsion points \([D]\) and \([E]\), can be calculated by the well-known formula \[ e_m ([D ],[ E])= \prod_P (-1 )^{m(\text{ord}_P D)(\text{ord}_P E)} {{g^{\text{ord}_P D}} \over {f^{\text{ord}_P E}}} (P), \] where \(P\) ranges over the geometric points of \(C\). We provide a new proof of this formula by reducing to the special case where \(k\) is finite. Our proof uses Kummer theory and class field theory to relate the Weil pairing to the Hilbert symbol, and in doing so explains the visual similarity between the formula above and Schmidt's explicit formula for the Hilbert symbol.
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 931965
zbMATH Keywords: curves over finite fields
zbMATH Keywords: Jacobian
zbMATH Keywords: torsion points
zbMATH Keywords: Weil pairing
zbMATH Keywords: Hilbert symbol
MaRDI profile type: Publication
cites work: Rational Points of Abelian Varieties Over Function Fields
cites work: Q3710637
description: scientific article; zbMATH DE number 931965