Distorting the volcano
This paper deals with ordinary elliptic curves over a finite field \(\mathbb{F}_q\) of characteristic \(p\) whose \(j\)-invariants are different from \(0\) and \(1728\). For a prime \(\ell\) with \(\ell\nmid q\) the volcanos of \(\ell\)-isogenies of such curves are considered. For a given elliptic curve \(E\) of a volcano of \(\ell\)-isogenies the question is how to determine the non-descending \(\ell\)-isogenies of \(E\). \textit{S. Ionica} and \textit{A. Joux}, [Math. Comput. 82, No. 281, 581--603 (2013; Zbl 1278.11067)] answered this question for a regular volcano or up to a second level of stability of an arbitrary volcano. In the paper under review the authors give an answer which works beyond the second stability level and does not need computing paths of \(\ell\)-isogenies. Denoting by \(D\) the discriminant of the order being isomorphic to the endomorphism ring of a given elliptic curve \(E\), they suppose \(\ell\neq 2,3, E[\ell]\subseteq E(\mathbb{F}_q)\), and \((\frac{D}{m})=1\), with \(m\neq \ell\). Then they consider a volcano of \(m\)-isogenies where \(E\) belongs to the crater and construct an endomorphism \(\varphi\) of \(E\) consisting of a cycle of \(m\)-isogenies of the crater and an isomorphism. They show: If \(P\) is a point of order \(\ell\) of \(E(\mathbb{F}_q)\) and \(\varphi\) is a distortion map for the subgroup \(\langle P\rangle\), then the \(\ell\)-isogeny with kernel \(\langle P\rangle\) is non-descending (Theorem 4.4). As a corollary they show that \(\varphi\) is a distortion map either for all kernels of the descending \(\ell\)-isogenies or for none of them. Further, they extend a result of \textit{D. Moody} (Proposition 5 in [Appl. Math. Comput. 218, No. 9, 5249--5258 (2012; Zbl 1251.14021)]) by comparing the crater size of volcanos of \(m\)-isogenies of two elliptic curves which belong to a given volcano of \(\ell\)-isogenies (Proposition 5.1). In addition, they present two algorithms: Taking the assumptions of Theorem 4.4 and the result of the Weil pairing \(W_\ell(\varphi(P),P)\) as the input, the first algorithm gives a sequence containing a generator of each kernel of the non-descending \(\ell\)-isogenies of \(E\). The input of the second algorithm is \(E/\mathbb{F}_q\), a prime \(m\), a point \(P\in E(\mathbb{F}_q)\) of order \(\ell\) such that the isogeny with kernel \(\langle P \rangle\) is descending. Its output is a sequence containing an ascending path from \(E\) and an indication whether one has reached the crater of the \(\ell\)-volcano. The paper ends with three examples: in the first two examples the two algorithms are applied and the third one presents an example for Proposition 5.1.
scientific article; zbMATH DE number 6810611
scientific article