Modular Equations

Abstract

The study of the algebraic-solutions problem for a second-order linear ordinary differential equation had brought to light the conceptual importance of considering groups of motions of the sphere, and, in particular, finite groups. Klein connected this study with that of the quintic equation, and so with the theory of transformations of elliptic functions and modular equations as considered by Hermite, Brioschi, and Kronecker around 1858. Klein’s approach to the modular equations was first to obtain a better understanding of the moduli, and this led him to the study of the upper half plane under the action of the group of two-by-two matrices with integer entries and determinant one; his great achievement was the production of a unified theory of modular functions. Independently of him, Dedekind also investigated these questions from the same standpoint, in response to a paper of Fuchs. So this chapter looks first at Fuchs’ study of elliptic integrals as a function of a parameter, and then at the work of Dedekind. The algebraic study of the modular equation is then discussed; the chapter concludes with Klein’s unification of these ideas.