The problem of classifying elliptic curves over $\mathbb Q$ with a given discriminant has received much attention. The analogous problem for genus $2$ curves has only been tackled when the absolute discriminant is a power of $2$. In this article, we classify genus $2$ curves C defined over ${\mathbb Q}$ with at least two rational Weierstrass points and whose absolute discriminant is an odd prime. In fact, we show that such a curve C must be isomorphic to a specialization of one of finitely many $1$-parameter families of genus $2$ curves. In particular, we provide genus $2$ analogues to Neumann–Setzer families of elliptic curves over the rationals.

Suppose $a^{2}(a^{2}+1)$ divides $b^{2}(b^{2}+1)$ with $b>a$. We improve a previous result and prove a gap principle, without any additional assumptions, namely $b\gg a(\log a)^{1/8}/(\log \log a)^{12}$. We also obtain $b\gg _{\unicode[STIX]{x1D716}}a^{15/14-\unicode[STIX]{x1D716}}$ under the abc conjecture.

We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval ℐ will contain asymptotically 2g∣ℐ∣ angles as the genus grows. We show that for the variance of number of angles in ℐ is asymptotically (2/π2)log (2g∣ℐ∣) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g∣ℐ∣ tends to infinity.

This series of papers presents and rigorously analyzes a probabilistic algorithm for finding small prime factors of an integer. The algorithm uses the Jacobian varieties of curves of genus 2 in the same way that the elliptic curve method uses elliptic curves. This second paper in the series is concerned with the order of the group of rational points on the Jacobian of a curve of genus 2 defined over a finite field. We prove a result on the distribution of these orders.

2000 Mathematical Subject Classification: 11Y05, 11G10, 11M20, 11N25.