The Spectrum of Rings

is called Riemann’s Theorem and guarantees the existence of functions whose poles can only occur at certain places. In fact, if D = ∑ ni(Pi) − ∑ mj(Qj), then H(D) consists of all functions f with zeros of order ≥ mj at Qj and poles of order ≤ ni at Pi. Riemann’s original motivation was to construct meromorphic functions on compact Riemann surfaces. For example, the meromorphic functions on a torus (which has genus 1) are exactly the elliptic functions. Riemann’s Theorem says that h(P ) ≥ 1, which means that the meromorphic functions with at most a single pole at P have dimension at least 1: actually, the only such functions are constants. But Riemann’s Theorem predicts h(2P ) ≥ 2, so there must exists nonconstant meromorphic functions whose only pole is at most a double pole at P : the only such functions are actually the Weierstrass ℘-functions. In order to get a better understanding of what is going on, let us look at the simplest case of a function field: the field of rational functions with coefficients from C, interpreted as the function field of the projective complex line. Note that the line has genus 0.