Constructing elliptic curves over finite fields with prescribed torsion

TL;DR

This paper introduces a local search algorithm to construct optimized equations for modular curves, enabling efficient generation of elliptic curves with prescribed torsion over finite fields.

Contribution

It develops a novel method for constructing equations of modular curves and applies it to generate elliptic curves with specific torsion properties efficiently.

Findings

Optimized equations for X_1(N) are constructed using a local search algorithm.

Efficient generation of elliptic curves with prescribed N-torsion over finite fields.

Fast methods for generating curves with points of order 4N using X_1(2N).

Abstract

We present a method for constructing optimized equations for the modular curve X_1(N) using a local search algorithm on a suitably defined graph of birationally equivalent plane curves. We then apply these equations over a finite field F_q to efficiently generate elliptic curves with nontrivial N-torsion by searching for affine points on X_1(N)(F_q), and we give a fast method for generating curves with (or without) a point of order 4N using X_1(2N).