There are many other proofs of finding all the possible automorphism groups of elliptic curves, but I am interested in the $Hint$ and the corresponding proof in the following exercise from Vakil's book.

Ex 19.10.E. Suppose $(E,e)$ is an elliptic curve over an algebraically closed field $k$ of characteristic not 2. Show that the automorphism group of $(E,e)$ is isomorphic to $\mathbb{Z}/2,\mathbb{Z}/4$, or $\mathbb{Z}/6$.

Hint: reduce to the question of automorphisms of $\mathbb{P}^1$ fixing the point $\infty$ and a set of distinct three points $\{p_1,p_2,p_3\} \in \mathbb{P}^1- \{\infty\}$.

So the proof is to use the fact that when $k=\bar{k}$ and $\text{char}~k \neq 2$, the elliptic curve is a double cover of $\mathbb{P}^1$ branched over $\{\infty,p_1,p_2,p_3\}$. So my question is how to show the 1-1 bijection between the automorphisms of $(E,e)$ and the automorphisms of $\mathbb{P}^1$ that fix $\{\infty\}$ and $\{p_1,p_2,p_3\}$, and what are they?


See Harthstorne, page 321, Corollary 4.7 and page 318, Lemma 4.4.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.