Hyperelliptic Curves

Elliptic curves can be generalized as follows.

A hyperelliptic curve $C$ of genus $g$ ($g \ge 1$) has the form:

\[ C: Y^2 + h(X) Y = f(X) \]

where $h$ is a polynomial with $\deg h \le g$, and $f$ is a monic polynomial with $\deg f \le 2g + 1$. Elliptic curves satisfy this definition for $g = 1$.

As for elliptic curves, each hyperelliptic curve contains a single point at infinity which we denote $O$. For a point $P = (x, y)$ on a hyperelliptic curve, let $\tilde P$ be the point $(x,-y - h(x))$. (So for elliptic curves, $\tilde P = -P$.) If $P = \tilde P$ then call $P$ a special point, otherwise call it ordinary. (On elliptic curves, special points are points of order 2.)

However, in general there is no group structure on the set of points of a hyperelliptic curve. Instead, we work on the jacobian group, which is defined below.

The Jacobian Group

Note that the definitions of function fields and divisors apply to any curve $C$. We define the jacobian group to be $Div^0(C) / Prin(C)$.

It turns out that each element of the jacobian group is equivalent to a divisor of the form

\[ \sum m_i P_i - (\sum m_i ) O \]

where $m_P \ne 0$ implies $m_{\tilde P} = 0$ unless $P$ is special in which case $m_P = 1$, and $\sum m_i \le g$. For elliptic curves, each element of the jacobian is equivalent to $P - O$ for some point $P$, and addition on points induces addition in the jacobian.