# Genus 2 Hyperelliptic curves

1. Sep 7, 2010

### Kenji Liew

Dear all,

I am currently doing my research deal with the Jacobian, Arithmetic and Pairings of the hyperelliptic curves ( for pure mathematics). I still at the stage of literature review. To enable me study the hyperelliptic curve, I will start with the elliptic curve and I need to read the book of number theory, algebraic geometry and group theory. However there are don't have many books on hyperelliptic curve.

Here are some problems about the hyperelliptic curve. I have came across of real and imaginary hyperelliptic curve but I cannot understand what the meaning of ordinary hyperelliptic curve and non-hyperelliptic curve?

I also wish to know what the better steps (materials) to be taken on doing the research of hyperelliptic curve in order to finish my postgraduate study. This is due to the scope is really wide.Thanks.

2. Sep 20, 2010

### mathwonk

an algebraic curve is often studied by its field of rational functions. A curve is hyperelliptic if that field is a quadratic extension of the rational function field K(t).

In geometric terms, a hyperelliptic curve has a degree two map onto the projective line P^1. All curves which do not are non hyperelliptic. Over the complexes say, all genus two curves are hyperelliptic, since the riemann roch theorem implies their canonical map is of degree two onto P^1. Hyperelliptic curves also exist in every positive genus, and for all genera g at least 2, the canonical map gives a degree two map to a copy of P^1 embedded in P^(g-1)

David Mumford's book Tata Lectures on Theta, book II, chapter 3, is an excellent source at least for hyperelliptic curves and their jacobians over the complex field.

3. Feb 28, 2011

### Voornaam

A hyperelliptic curve admits a double covering map onto the projective line, i.e., the Riemann sphere if one works over the complex numbers. This map is usually called the canonical map. Therefore, a curve is hyperelliptic if it is not isomorphic onto its image under the canonical map. Meaning, under this map it is not a projective embedding. Non-hyperelliptic curves are a projective embedding under the canonical map. An Euler characteristic computation (Riemann Hurwitz formula) shows that any hyperelliptic curve of genus g \geq 2 is ramified at 2g+2 points. Since we know that there exists an action of PSL(2,C) acting on P^1 by (x,y,z) -> (0,1,\infty) it follows that any hyperelliptic curve is of the form x(x-1)(x-a_i) where i=1,...,2g-1. To distinguish between (I always forgot about the names) complex and real hyperelliptic curves is simple to distinguish between whether or not roots of its equation lying at infinity. However, I'am doing research to low genus hyperelliptic curves; so be patiented and a good reference will appear soon :P