Counting hyperelliptic curves

We find a closed formula for the number hyp (g) of hyperelliptic curves of genus g over a finite field k = Fq of odd characteristic. These numbers hyp (g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q - 1 and q + 1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is defined to be self-dual if it is k-isomorphic to its own hyperelliptic twist. © 2009 Elsevier Inc. All rights reserved.