Elliptic 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 $\mathrm{deg}h\le g$, and $f$ is a monic polynomial with $\mathrm{deg}f\le 2g+1$. 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 ${\mathrm{Div}}^0(C)/\mathrm{Prin}(C)$.

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.