We classify minimal surfaces of general type with ${{p}_{g}}=q=2$ and ${{K}^{2}}=6$ whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth irreducible components ${{\mathcal{M}}_{Ia}},\,{{\mathcal{M}}_{Ib}},\,{{\mathcal{M}}_{II}}$ of dimension 4, 4, 3, respectively.

We study, from the point of view of abelian and Kummer surfaces and their moduli, the special quintic threefold known as Nieto's quintic. It is, in some ways, analogous to the Segre cubic and the Burkhardt quartic and can be interpreted as a moduli space of certain Kummer surfaces. It contains 30 planes and has 10 singular points: we describe how some of these arise from bielliptic and product abelian surfaces and their Kummer surfaces.

Let $V$ be a $K3$ surface defined over a number field $k$. The Batyrev-Manin conjecture for $V$ states that for every nonempty open subset $U$ of $V$, there exists a finite set ${{Z}_{U}}$ of accumulating rational curves such that the density of rational points on $U\,-\,{{Z}_{U}}$ is strictly less than the density of rational points on ${{Z}_{U}}$. Thus, the set of rational points of $V$ conjecturally admits a stratification corresponding to the sets ${{Z}_{U}}$ for successively smaller sets $U$.

In this paper, in the case that $V$ is a Kummer surface, we prove that the Batyrev-Manin conjecture for $V$ can be reduced to the Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$ induced by multiplication by $m$ on the associated abelian surface $A$. As an application, we use this to show that given some restrictions on $A$, the set of rational points of $V$ which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Batyrev-Manin type.

We prove that if q is a power of an odd prime, then there is no genus-2 curve over $\mathbf{F}_q$ whose Jacobian has characteristic polynomial of Frobenius equal to $x^4 + (2 - 2q)x^2 + q^2$. Our proof uses the Brauer relations in a biquadratic extension of $\mathbb{Q}$ to show that every principally polarized abelian surface over $\mathbf{F}_q$ with the given characteristic polynomial splits over $\mathbf{F}_{q^2}$ as a product of polarized elliptic curves.

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.