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)?
 A: 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.
A: Felipe's answer is much better, but what about this, just for fun:  an automorphism f of X induces an automorphism of effective divisors of degree 2, i.e. of the symmetric square X^(2).  This object contains exactly one copy of P^1, corresponding to the divisors of the g(1,2) and thus the automorphism f^(2) of X^(2) induces an automorphism of this P^1, which takes divisors of form 2.P to divisors of form 2.f(P).
A: 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.
