The Prime Form E(x,y).

Abstract

Given an arbitrary compact Riemann surface X, of genus g, wouldn’t it be handy if we had a holomorphic function E: X × X → ¢ such that E(x,y) = 0 if and only if x = y? Although such a function doesn’t exist, it turns out that it “almost” does! To understand part of the problem and how to fix it, let’s look at the simplest case: Example. Let X = IP¹. The function x-y works on IP¹-{∞} but not on all of IP¹. So consider instead the “differential”:

$$ {\rm E}\left( {x,y} \right) = \frac{{x - y}} {{\sqrt {dx} \sqrt {dy} }}, $$

where $\sqrt {dx} ,\sqrt {dy}$ are defined as follows.