A **hyperelliptic curve** $X$ over a field $k$ is a smooth projective algebraic curve of genus $g\ge 2$ that admits a 2-to-1 map $X\to \mathbb{P}^1$ defined over the algebraic closure $\bar k$.

Over $k$ there is always a 2-to-1 map from $X$ to a genus 0 curve $C$, but the curve $C$ need not be isomorphic to $\mathbb{P}^1$; this holds if and only if $C$ has a $k$-rational point.

If $X$ admits a 2-to-1 map to $\mathbb{P}^1$ that is defined over $k$, then $X$ has a **Weierstrass model** of the form $y^2+h(x)y=f(x)$; when the characteristic of $k$ is not $2$ one can complete the square to put this model in the form $y^2=f(x)$.

In general, there is always a model for $X$ in $\mathbb{P}^3$ of the form
\[
h(x,y,z)=0\qquad w^2=f(x,y,z)
\]
where $h(x,y,z)$ is a homogeneous polynomial of degree $2$ (a **conic**) and $f(x,y,z)$ is a homogeneous polynomial of degree $g+1$.

**Authors:**