Monodromy and normal forms

TL;DR

This paper explores the historical development of monodromy theory, compares elliptic curve normal forms through their monodromy groups, and discusses the emergence of Jacobian varieties and complex analysis methods.

Contribution

It provides a historical and geometric comparison of elliptic curve normal forms based on monodromy and stabilizer groups, and discusses the origins of Jacobian varieties and Abelian functions.

Findings

Comparison of Weierstraf, Legendre, and other normal forms via monodromy groups

Explanation of geometric meaning and stabilizer groups in P SL(2,Z)

Illustration of complex analysis methods in difference equations

Abstract

We discuss the history of the monodromy theorem, starting from Weierstra\ss, and the concept of monodromy group. From this viewpoint we compare then the Weierstra\ss , the Legendre and other normal forms for elliptic curves, explaining their geometric meaning and distinguishing them by their stabilizer in P SL(2,Z) and their monodromy. Then we focus on the birth of the concept of the Jacobian variety, and the geometrization of the theory of Abelian functions and integrals. We end illustrating the methods of complex analysis in the simplest issue, the difference equation $f(z) = g(z+1) - g(z)$ on $\mathbb C$.