## Hyperelliptic Curves

Elliptic curves can be generalized as follows.

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

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

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.

*blynn@cs.stanford.edu*💡