Elliptic and Abelian Functions

Riemann never published work dealing directly with elliptic functions, but they played a major role in his lecture courses and in many ways served to motivate his general theory of Abelian functions. For this reason, not to mention their central place in the emergence of complex analysis, Riemann's treatment of the genus one case of Abelian functions is given due attention in this chapter. Elliptic functions first arose when Abel and Jacobi realized that the inversion of an elliptic integral leads to a double-periodic function, studied by Riemann using the Gaussian plane of complex numbers. The next higher case -- a so-called hyperelliptic integral of genus two -- led Jacobi to ask whether a comparable inversion theorem could also be found. He then showed that it could, if one took the sum of two definite integrals and defined the inverse functions for the sum and product of their upper endpoints in terms of the two given integral values. This resulted in single-valued functions with four periods, prompting Jacobi to pose the more general inversion problem. This required inverting a system of p hyperelliptic integrals of genus p. Weierstrass later solved that celebrated problem when he was still a Gymnasium teacher in a remote part of Germany. After gaining a professorship in Berlin, he went on to become the most influential mathematician of the era. Naturally, he wanted to attack the even more demanding inversion problem, which arises from a system of integrals based on an algebraic function of given genus, which would yield a large class of Abelian functions. While working on this, however, Riemann published his general solution in 1857, a startling result that caused Weierstrass to rethink his whole approach. The basic arguments Riemann laid out in his famous paper on Abelian functions constitute the centerpiece of the present chapter.