On the $j$-invariant and the Legendre Representation of Elliptic Curves with Complex Multiplication

TL;DR

This paper introduces a novel method for determining the Legendre form of elliptic curves with complex multiplication using meromorphic functions, avoiding Hilbert class polynomial computations, and provides algorithms for computing special modular function values.

Contribution

A new approach using meromorphic functions with specific ramification to find Legendre representations of CM elliptic curves, bypassing traditional Hilbert class polynomial methods.

Findings

Explicit polynomial systems for Legendre representations

Algorithm for computing $j(k\tau)$ from $j(\tau)$ for $k=2,3$

Method for calculating special values of the modular function $j$

Abstract

By introducing a class of meromorphic functions with certain ramification structures on $\Bbb{CP}^1$, a new method for the determination of the Legendre representation of elliptic curves with complex multiplication is introduced. These functions reduce the desired representation to the solution of an explicitly given system of polynomial equations and makes no use of the knowledge of Hilbert class polynomial on which standard computations depend. As a byproduct, an algorithm for computing $j(k\tau)$ in terms of $j(\tau)$ is obtained and implemented for $k=2,3$. Solving the system of equations provides a method for computing certain special values of the modular function $j:\Bbb{H}\rightarrow\Bbb{C}$.