# Reference for hyperelliptic curves

I was reading a paper the other day that said that all automorphisms of a hyperelliptic curve are liftings of automorphisms of $\mathbb{P}^1$ operating on the set of branch points.

Can someone point me to a good reference for hyperelliptic curves that would explain that (and possibly other important things about them too)?

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According to Weil (projecteuclid.org/…) Ch IV, §9 of Chevalley's book "Introduction to the theory of algebraic functions of one variable", is a "tour de force". –  Felipe Voloch Aug 13 '13 at 22:31
@FelipeVoloch "a tour de force" is usually not an indication of "a good reference". –  Igor Rivin Aug 14 '13 at 2:07
Mumford's Tata lectures on theta has a chapter on hyperelliptic curves. (I remember someone called Chevalley's book "inhuman". I tend to agree with this:-) –  Alexandre Eremenko Aug 14 '13 at 2:24
@AlexandreEremenko Yes it is Weil who calls it "severely dehumanized", in the above-linked review. –  Francois Ziegler Aug 14 '13 at 2:48

Joking aside... Let $x: X \to \mathbb{P}^1$ be of degree $2$ and $\sigma: X \to X$ be an automorphism. Consider $f: X \to \mathbb{P}^1\times\mathbb{P}^1, f=(x,x\circ\sigma)$. If $f$ is injective, then $X$ embeds as a curve of bidegree $(2,2)$ in $\mathbb{P}^1\times\mathbb{P}^1$ and therefore has genus at most $1$. So, if $X$ has genus $>1$ then $f$ is $2-1$ and $f(X)$ has bidegree $(1,1)$ in $\mathbb{P}^1\times\mathbb{P}^1$, so $f(X)$ is the graph of an automorphism $\tau$ of $\mathbb{P}^1$ and $x\circ\sigma = \tau \circ x$. Finally $\tau$ has to send branch points to branch points because $\sigma$ sends ramification points to ramification points, as these points are Weierstrass points, and automorphisms preserve Weierstrass points.
The previous two answers are really good; another way to see it is the following: Using Riemann-Roch one can see that the $2:1$ morphism $f:X\to\mathbb{P}^1$ is unique modulo automorphisms of $\mathbb{P}^1$. If $\sigma:X\to X$ is an automorphism of $X$ then $f\circ\sigma$ is a $2:1$ morphism from $X$ to $\mathbb{P}^1$ and so there must be an automorphism $\eta:\mathbb{P}^1\to\mathbb{P}^1$ such that $f\circ\sigma=\eta\circ f$. This automorphism clearly sends branch points to branch points.