Given a hyperelliptic curve defined over a finite field, there is a quantity called the a-number attached to the curve. This integer ranges from 0 to the genus and curves with a-number 0 are called "ordinary." For example, elliptic curves have genus 1 and those curves with a-number 1 are the supersingular elliptic curves ("extraordinary"). Enumerating hyperelliptic curves with given genus and/or a-number can be accomplished with more modern computing, and this leads one to conjecture probabilities for curves of given genus to have a particular a-number.

In this talk we'll discuss finite fields, function fields, what it means to be a hyperelliptic curve over a finite field, and how one computes the genus and a-number. We'll recall previous work regarding the probabilities above and some methods used to address them along with some data indicating these methods may be lacking. We'll then demonstrate a relatively simple counting method, based on the theory of heights in Diophantine geometry, that definitively answers the question of how many hyperelliptic curves of given genus have a particular a-number in the case of characteristic 3.

This is joint work with Derek Garton and Colin Weir.