K3 surfaces over finite fields with given L-function

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...

Potential Density of Rational Points for K3 Surfaces over Function Fields

We give examples of non-isotrivial K3 surfaces over complex function fields with Zariski-dense rational points and NéronSeveri rank one.

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.