Weil Differentials and the Canonical Class

  • Michael Rosen
Part of the Graduate Texts in Mathematics book series (GTM, volume 210)


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.


Meromorphic Function Function Field Compact Riemann Surface Divisor Class Discrete Valuation Ring 
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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Michael Rosen
    • 1
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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