Traces of division polynomials in characteristic two
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 \(\varepsilon_1(J\circ I)=\varepsilon_1(I)+\varepsilon_1(J)\).
scientific article; zbMATH DE number 1598956