What is the maximum number of ramification points that a mapping of finite degree from one smooth projective curve over \(\CC\) of genus 1 to another smooth projective curve of genus 1 can have? Give an explanation for your answer.
Solution
Let \(f\colon X\to Y\) be a finite morphism of degree \(d\) between smooth irreducible projective curves over \(\CC\) of genus \(1\), and let \(e_p\geq 1\) be the ramification index of \(f\) at \(p\in X\).
1. \(\sum_{p\in X}(e_p-1)=0\).
The Riemann–Hurwitz formula gives \[ 2g(X)-2 = d\,\bigl(2g(Y)-2\bigr) + \sum_{p\in X}(e_p-1). \] With \(g(X)=g(Y)=1\) both sides of \(2g-2\) vanish, so the sum is \(0\).
2. Q.E.D.
Each term \(e_p-1\) is nonnegative, so step 1 forces \(e_p=1\) for every \(p\). The maximum number of ramification points is \(\boxed{0}\): every such map is unramified.