Abstract:

It is shown that automorphism group of any Riemann surface X of genus g > or = 2 is finite. Also given a bound to the cardinality of the automorphism group, depending on the genus, speci fically
Aut(X) < or = 84(g1). This bound will be obtained by applying Hurwitz formula to the natural holomorphic map from a Riemann surface to it's quotient under action of the finite group Aut(X). The finiteness
is proved by considering a homomorphism from Aut(X) to the permutation group of a finite set and showing that the kernel is finite. The finite set under consideration is the set of Weierstass points. p is a Weierstass point, if the set of integers n, such that there is no f {element of} M(X) whose only pole is p with order n, is not {1, ... g}. All these are explained in Chapter 4. RiemannRoch Theorem is heavily
used which is proved in Chapter 3. Proof of RiemannRoch Theorem requires existence of nonconstant meromorphic functions on a Riemann surface, which is proved in Chapter 2. Basics are dealt with in Chapter 1. 