Equivalence of two canonical lists of elliptic functions

TL;DR

This paper proves a correspondence between four types of elliptic solutions to certain differential equations and four specific binomial equations, clarifying their equivalence.

Contribution

It establishes a one-to-one correspondence between elliptic solutions of Schwarzian differential equations and Briot-Bouquet binomial equations.

Findings

Identifies four types of elliptic solutions for Schwarzian equations

Shows equivalence with four Briot-Bouquet binomial equations

Provides a classification linking elliptic functions and differential equations

Abstract

We establish a one-to-one correspondence between,on one hand the four types of transcendental meromorphic solutions of the autonomous Schwarzian differential equations which are elliptic,on the other hand the four binomial equations of Briot and Bouquet possessing an elliptic solution.