## 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.

Ben Lynn blynn@cs.stanford.edu 💡