Traces of division polynomials in characteristic two
type: Item
instance of: scholarly article
title: Traces of division polynomials in characteristic two
zbMATH Open document ID: 0972.11050
author: Antonia Wilson Bluher
published in: Journal of Number Theory
publication date: 2001-05-20T00:00:00Z
reviewed by: Amílcar Pacheco
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 1598956
zbMATH Keywords: division polynomials
zbMATH Keywords: isogenies of elliptic curves
MaRDI profile type: Publication
full work available at URL: jnth.2000.2625
OpenAlex ID: W2064199632
cites work: Relations between certain power sums of elliptic modular forms in characteristic two
cites work: Q3840191
cites work: Q3710637
DOI: 10.1006/JNTH.2000.2625
review text: Let \(E\) be an ordinary elliptic curve defined over a perfect field \(F\) of characteristic two. For each positive odd integer \(\ell\) the \(\ell\)-division polynomial is defined by \(\psi_{\ell}=\prod_{i=1}^{(\ell^2-1)/2}(x-x_i)\), where the \(x_i\)'s are the distinct \(x\)-coordinates of the non-zero points of the \(\ell\)-torsion subgroup \(E[\ell]\) of \(E\). The first result of this paper concerns the calculation of the trace of \(\psi_{\ell}\), which at first would be in \(F\), but it is proved that it actually lies in \(\mathbb F_2\) and its value is determined by the Jacobi symbol \((\frac 2{\ell})\). NEWLINENEWLINENEWLINEThe second result deals with additivity properties of isogenies. More precisely, let \(I:E\to E'\) be an isogeny of elliptic curves of odd degree. Denote by \(\varepsilon_1(I)\) the sum of the distinct \(x\)-coordinates of \(\ker(I)\). If \(I:E\to E'\) and \(J:E'\to E''\) are two isogenies of elliptic curves of odd degree, then \(\varepsilon_1(J\circ I)=\varepsilon_1(I)+\varepsilon_1(J)\).
description: scientific article; zbMATH DE number 1598956