Holomorphic functions on algebraic curves

Question

I have been asked to solve the following problem, but I really need some help...

How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1?

I know that I have to use Hurwitz' Formula, but I didn't understand it at all. Thanks!

Answer 1

The Riemann–Hurwitz formula says that if $f: C \rightarrow D$ is a finite holomorphic map of degree $d$ between any two algebraic curves, then

$$\chi(C) = d \cdot \chi(D) - \sum_{P \in C} (e_P-1)$$

where $e_P$ is the so-called ramification index of $f$ at the point $P \in C$. (This is a natural number with the property that $e_P=1$ if and only if the map $f$ is not ramified at $P$. Note that $e_P=1$ for all but finitely many points $P$, so the sum here really makes sense.)

In your case, $C$ and $D$ both have genus 1, hence $\chi=2-2g=0$ on both sides of the equation: this means that we must have $e_P=1$ for all points $P \in C$. In other words, the map $f$ must be unramified everywhere.

Answer 2

Adding to Asal Beag Dubh's answer, use a little bit of topology (for example, $\mathbb{C}$ is the universal covering space of both curves) to show that your holomorphic map lifts to a linear map on $\mathbb{C}$, translated by a constant. In other words, if $f$ sends 0 to 0 (I'm assuming you know that a genus 1 curve is a group), it is a group homomorphism.