Constructing elliptic curves over finite fields using double eta-quotients

Abstract

We examine a class of modular functions for Γ0(N) whose values generate ring class fields of imaginary quadratic orders. This fact leads to a new algorithm for constructing elliptic curves with complex multiplication. The difficulties arising when the genus of X0(N) is not zero are overcome by computing certain modular polynomials. Being a product of four η-functions, the proposed modular functions can be viewed as a natural generalisation of the functions examined by Weber and usually employed to construct CM-curves. Unlike the Weber functions, the values of the examined functions generate any ring class field of an imaginary quadratic order regardless of the congruences modulo powers of 2 and 3 satisfied by the discriminant.