Differential equations and group theory from Riemann to Poincare
The origins of the theory of modular and automorphic functions are found in the work of Legendre, Gauss, Jacobi, and Kummer on elliptic functions and the hypergeometric equation. Riemann's work on this differential equation gave a decisive impulse to the global theory of the solutions to such equations, and was extended by Fuchs who raised the problem: when are all solutions to a linear differential equation algebraic? This problem was tackled in various ways by Schwarz, Fuchs himself, Gordan, Jordan, and Klein, with results that displayed the new methods of group theory to advantage. At the same time, or a little earlier, the theory of modular transformations proclaimed by Galois was explored by several mathematicians, notably Hermite, and Klein was able to unite that work with his geometrical methods and the crucial observations of Dedekind. This work marks the origin of the Galois theory of function fields and the systematic study of modular functions. The theory of linear differential equations was then further extended by Poincare, who brought to it geometric and group-theoretic insights strikingly similar to, but at first independent of, those of Klein, and who opened-up the theory of automorphic functions.