N-systems, class polynomials for double eta-quotients and singular values of J-invariant function

TL;DR

This paper improves the efficiency of constructing elliptic curves over finite fields by analyzing modular invariants and providing conditions to reduce computational complexity in counting rational points.

Contribution

It introduces a new condition for modular invariants as multiple roots of modular polynomials, reducing the computational effort in elliptic curve point counting.

Findings

Identifies when modular invariants are multiple roots of modular polynomials.

Provides a method to decrease the computational load in point counting.

Enhances the process of constructing elliptic curves using double eta-quotients.

Abstract

Enge and Schertz gave the method of using the double eta-quotient for the construction of elliptic curves over finite fields. In their method, it is necessary to count the number of rational points of elliptic curves corresponding to solutions of the modular equation over a finite field, because in advance we can not know which solution of the modular equation is that corresponding to the modular invariant. We give a condition that the modular invariant is a multiple root of the modular polynomial. Consequently, we give a method to reduce the amount of computation in the process of counting the number of rational points.