Chapter IX. The Number Theory of Algebraic Curves

Abstract

This chapter investigates algebraic curves from the point of view of their function fields, using methods analogous to those used in studying algebraic number fields.

Section 1 gives an overview, explaining how Riemann's theory of Riemann surfaces of functions ties in with the notion of an algebraic curve and explaining how such curves can be investigated through the discrete valuations of their function fields. It is shown that what needs to be studied is arbitrary function fields in one variable over a base field. It is known that every compact Riemann surface can be viewed as an algebraic curve irreducible over $\mathbb{C}$, and thus the function fields of compact Riemann surfaces are to be viewed as informative examples of the theory in the chapter.

Section 2 introduces the notion of a divisor, which is any formal finite $\mathbb{Z}$ linear combination of the discrete valuations of the function field that are trivial on the base field, and the notion of the degree of a divisor, which is the sum of its coefficients weighted suitably. Each nonzero member $x$ of the function field gives rise to a principal divisor $(x)$, and the main result of the section is that the degree of every principal divisor is 0. This is an analog for function fields of the Artin product formula for number fields.

Section 3 contains the definition of the genus of the function field under study. The main object of study is the vector space $L(A)$ for a divisor $A$; this consists of 0 and all nonzero members $x$ of the function field such that $(x)+A$ is a divisor $\geq0$. Roughly speaking, it may be viewed as the space of functions on the zero locus of the curve whose poles are limited to finitely many points and to a certain order depending on the point. The genus is defined in terms of $\dim L(A)-\deg A$ when $A$ is a divisor that is a large multiple of the pole part of any fixed principal divisor. The main result of the section is Riemann's inequality, which says that $\dim L(A)\geq \deg A+1-g$ for all divisors $A$, where $g\geq0$ is the genus, and that $g$ is the smallest integer that works in this inequality for all divisors $A$.

Sections 4–5 concern the Riemann–Roch Theorem, which gives an interpretation of the difference of the two sides of Riemann's inequality as $\dim L(B)$ for a suitable divisor $B$ that can be defined in terms of $A$. Section 4 gives the statement and proof of the theorem, and Section 5 gives a number of simple applications.