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 $\bar k$.

If $X$ is a hyperelliptic curve over $k$, then the canonical map $X \to \mathbb{P}^{g-1}$ is a 2-to-1 map onto a smooth genus 0 curve $Y$. The curve $Y$ is isomorphic to $\mathbb{P}^1$ if and only if $Y$ 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:**

**Knowl status:**

- Review status: beta
- Last edited by Bjorn Poonen on 2022-03-24 16:23:46

**Referred to by:**

- ag.cluster_picture
- av.hyperelliptic_count
- columns.av_fq_isog.hyp_count
- g2c.all_rational_points
- g2c.g2curve
- g2c.minimal_equation
- g2c.num_rat_wpts
- g2c.simple_equation
- g2c.two_torsion_field
- lmfdb/higher_genus_w_automorphisms/main.py (line 1287)
- lmfdb/higher_genus_w_automorphisms/templates/hgcwa-index.html (line 48)
- lmfdb/higher_genus_w_automorphisms/templates/hgcwa-show-passport.html (line 44)

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