Compact Riemann Surfaces

Abstract

This big chapter is devoted to the theory of compact Riemann surfaces. Tori are examples of compact Riemann surfaces. This means that we generalize the theory of elliptic functions here. A compact Riemann surface can be associated with any algebraic function, and in this way we obtain all compact Riemann surfaces. The compact Riemann surfaces achieve the same result for the integration of algebraic functions as does the theory of elliptic functions for the elliptic integrals. The triumph of the theory of Riemann surfaces was that it made the “integrals of the first kind” understandable and solved the so-called Jacobi inversion problem. We have to go a long way to achieve this aim. At the end, we shall arrive at the best-known theorems of the theory of Riemann surfaces, such as the the Riemann–Roch theorem, Abel’s theorem, and the Jacobi inversion theorem. On the way, we must also understand the topology of compact Riemann surfaces. We shall treat the topological classification completely here.