# Principal divisors on a compact Riemann surface

Let $$X$$ be a compact Riemann surface, and $$f$$ a meromorphic function on X. There's a theorem telling us that $$\deg(\mathrm{div}(f)) = 0$$.

But is the inverse statement also true? I mean, is it true that:

if $$D$$ is a divisor on $$X$$ with $$\deg(D) = 0$$, then exists a meromorphic function $$f$$ on $$X$$ such that $$D = \mathrm{div}(f)$$?

Thanks!

• I'm quite sure the answer is no... for example, there is no function on the complex torus with only one simple zero and one simple pole. Nov 4, 2011 at 20:41
• (degree-$0$ divisors)/(principal divisors) is the Jacobian variety of your Riemann surface. It is a complex torus of (complex) dimension $g$ (= the genus of the surface). See e.g. en.wikipedia.org/wiki/Abel%E2%80%93Jacobi_map Nov 4, 2011 at 21:42

That is not true. Here a counterexample: consider a Riemann surface $X$ of genus $g \geq 1$. Fix $p, q \in X$ distinct points and consider the divisor $D = p - q$. This divisor has degree $0$, but it is not principal, because on the contrary there would be a holomorphic map $f: X \rightarrow \overline{\mathbb{C}}$ of degree equal to $1$ (for it has single simple zero/infinity value), and it is well known that a such map with this property is an isomorphism. That is a contradiction, since $g(\overline{\mathbb{C}}) = 0$. Look for Abel-Jacobi Theorem for necessary and sufficient conditions for a divisor be principal.
Rafael answer shows that if every degree zero divisor on $$X$$ is principal then $$X$$ is isomorphic to the Riemann sphere $$\mathbb{C}_{\infty}$$.
The converse also holds true, i.e., every degree zero divisor $$D$$ on $$\mathbb{C}_{\infty}$$ is principal. To see this just note that if $$D = \sum n_i \cdot z_i + n_{\infty}\cdot \infty$$ and $$n_\infty = - \sum n_i$$ then $$f(z) = \Pi (z-z_i)^{n_i}$$ is a rational function (hence meromorphic on $$\mathbb{C}_{\infty}$$) such that div$$(f) = D$$.
• Shouldn't it be " if every degree 0 divisor on $X$ is principal, then $X\simeq\mathbb P^1$''? Nov 20, 2020 at 13:29