Weil Differentials and the Canonical Class

Abstract

In the last chapter we gave some definitions and then the statement of the Riemann-Roch theorem for a function field K/F. In this chapter we will provide a proof. In the statement of the theorem an integer, g,enters which is called the genus of K. Also, a divisor class, ℂ, makes an appearance, the canonical class of K. We will provide another interpretation of these concepts in terms of differentials. Thus, differentials give us the tools we need for the proof and, as well, lead to a deeper understanding of the theorem. In addition, the use of differentials will enable us to prove two important results: the strong approximation theorem and the Riemann-Hurwitz formula. The first of these will be proven in this chapter, the second in Chapter 7, where we will also prove the ABC conjecture in function fields and give some of its applications.