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 💡