Using the Charlap-Coley-Robbins polynomials for computing isogenies

Abstract

The SEA algorithm for computing the cardinality of elliptic curves over finite fields in many characteristic uses modular polynomials. These polynomials come into different flavors, and methods to compute them flourished. Once equipped with some modular polynomials for prime $\ell$, algebraic formulas are used to compute a curve $\mathcal{E}^*/\mathbb{F}_q$ that is $\ell$-isogenous to the curve of interest $\mathcal{E}$. These formulas involve derivatives of the modular polynomial that may sometime vanish. One way to overcome this problem is to use alternative trivariate polynomials $U_\ell$, $V_\ell$, and $W_\ell$ introduced by Charlap, Coley and Robbins to overcome some difficulties in the first versions of Elkies's approach. We give properties of these polynomials, as well as formulas to compute the isogenous curve that were sketched by Atkin. Also we investigate another suggestion of Atkin using modular polynomials associated to a power product of Dedekind's $\eta$ function.