Counting hyperelliptic curves (Q1018415)

Let \(k\) be the finite field of odd order \(q\). In this paper a closed formula for the cardinality of the set \(\text{Hyp}(g)\) of \(k\)-isomorphism classes of hyperelliptic curves over \(k\) of genus \(g\) is found. The formula is expressed as a polynomial in \(q\) with integers coefficients that depend on \(g\) and the divisors of \(q-1\) and \(q+1\), and indeed it is asymptotically \(2q^{2g-1}\) being \(q^{2g-1}\) the number of \(k\)-rational points of \(\text{Hyp}(g)\). In addition, a closed formula for the number of self-dual curves of genus \(g\) is obtained. A hyperelliptic curve over \(k\) is called self-dual if it is \(k\)-isomorphic to its own hyperelliptic twist.