I'm self-studying Silverman and Tate's Rational Points on Elliptic Curves, and I came across this exercise:
Exercise 6.3:
(a) Suppose that $\lambda(z)=a_0z^n+a_1z^{n-1}+...+a_{n-1}z+a_n$ is a polynomial of degree $n$ such that $\lambda:\mathbb C^*\rightarrow \mathbb C^*$ is a homomorphism. Prove that $\lambda(z)=z^n$.
(b) Suppose that $\lambda(z)$ is a meromorphic function such that $\lambda:\mathbb C^*\rightarrow \mathbb C^*$ is a homomorphism. Prove that $\lambda(z)=z^n$ for some $n\in \mathbb Z$.
This book doesn't depend on much complex analysis, so I was initially thrown off and tried to solve this with elliptic curves/Galois groups. I considered using Rouche's theorem or Liouville's theorem, but Rouche's theorem only gives the number of zeros (which is already clear through the Fundamental Theorem of Algebra), and Liouville's theorem implies $z^n$ would be constant if I could find a bound for the polynomial $\lambda(z)$. I also tried using Little/Big Picard (which seems the most promising), but I'm stumped on "narrowing down" $\lambda(z)$ to a monic, single-term polynomial.
I have two main questions:
- I'm not sure how to prove the statements, as the standard theorems I've looked at do not fully satisfy the claim. I'm thinking of different ways to bound/modify $\lambda(z)$ such that it behaves closer to $z^n$ as $z\rightarrow \infty$, but that doesn't solve the issue of local behavior (especially around $z=0$)
- The focus of the chapter was on complex multiplication, abelian extensions over the rationals, and how elliptic curves were used to prove Fermat's last theorem. How does this question relate to the content of the chapter? Theorem 6.16 states that a rational map that preserves the identity element/projective point at infinity (in the group law) between two elliptic curves is a homomorphism – is this exercise meant to bridge a gap between homomorphisms between elliptic curves and general polynomials, or address some other aspect of rational functions?
If $\lambda$ had any non-zero root, then $\lambda$ would not even be a map from $\Bbb C^\times$ to $\Bbb C^\times$.
Same with poles.