Large Orbits on Markoff-Type K3 Surfaces over Finite Fields

K3 surfaces over number fields with geometric Picard number one

A long-standing question in the theory of rational points of algebraic surfaces is whether a K3 surface X over a number field K acquires a Zariski-dense set of L-rational points over some finite extension L/K. In this case, we say X has potential density of rational points. In case XC has Picard rank greater than 1, Bogomolov and Tschinkel [2] have shown in many cases that X has potential densi...

Abelian Surfaces over Finite Fields as Jacobians

Counting Elliptic Surfaces over Finite Fields

We count the number of isomorphism classes of elliptic curves of given height d over the field of rational functions in one variable over the finite field of q elements. We also estimate the number of isomorphism classes of elliptic surfaces over the projective line, which have a polarization of relative degree 3. This leads to an upper bound for the average 3-Selmer rank of the aforementionned...

Surfaces in P over finite fields

For a smooth surface X in P of degree d, defined over a finite field Fq with q elements, q prime, we prove that X has at most d(d+q−1)(d+2q−2)/6+d(11d−24)(q+1) points with coordinates in Fq.

Expansion of Orbits of Some Dynamical Systems over Finite Fields

Given a finite field Fp = {0, . . . , p − 1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ 7→ ψ(ξ) associated with a rational function ψ ∈ Fp(X). We use bounds of exponential sums to show that if N ≥ p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the...