Algebraic Function Fields and Global Function Fields

Abstract

So far we have been working with the polynomial ring A inside the rational function field k = F(T). In this section we extend our considerations to more general function fields of transcendence degree one over a general constant field. This process is somewhat like passing from elementary number theory to algebraic number theory. The Riemann-Roch theorem is the fundamental result needed to accomplish this generalization. We will give a proof of this fundamental result in Chapter 6. In this chapter we give the basic definitions, state the theorem, and derive a number of important corollaries. After this is accomplished, attention will be shifted to function fields over a finite constant field. Such fields are called global function fields. The other class of global fields are algebraic number fields. All global fields share a great number of common features. We introduce the zeta function of a global function field and explore its properties. The Riemann hypothesis for such zeta functions will be explained in some detail, and we will derive several very important consequences, among others an analogue for the prime number theorem for arbitrary global function fields. A proof of the Riemann hypothesis will be given in the appendix. In this chapter we will prove a weak version. This is enough to yield the analogue of the prime number theorem, albeit with a poor error term. In later chapters we will also explore L-functions associated to global function fields - both Hecke L-functions (generalizations of Dirichlet L-functions) and Artin L-functions.