Rational points on elliptic K3 surfaces of quadratic twist type

In studying rational points on elliptic K3 surfaces of the form $f(t)y^2=g(x)$, where $f,g$ are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell-Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels-Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.Comment: v3 v4 Completely rewritten. Results strengthened. v5 some footnotes added, published versio

Dynamics on supersingular K3 surfaces

For any odd characteristic p=2 mod 3, we exhibit an explicit automorphism on the supersingular K3 surface of Artin invariant one which does not lift to any characteristic zero model. Our construction builds on elliptic fibrations to produce a closed formula for the automorphism's characteristic polynomial on second cohomology, which turns out to be an irreducible Salem polynomial of degree 22 with coefficients varying with p.Comment: 12 pages, 3 figures; v2: main result improved to Salem degree 2

Distribution of Mordell--Weil ranks of families of elliptic curves

We discuss the distribution of Mordell--Weil ranks of the family of elliptic curves $y^2=(x+\alpha f^2)(x+\beta b g^2)(x+\gamma h^2)$ where $f,g,h$ are coprime polynomials that parametrize the projective smooth conic $a^2+b^2=c^2$ and $\alpha,\beta,\gamma$ are elements from $\overline{\mathbb{Q}}$. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.Comment: 28 page

Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19

It is known that K3 surfaces S whose Picard number rho (= rank of the Neron-Severi group of S) is at least 19 are parametrized by modular curves X, and these modular curves X include various Shimura modular curves associated with congruence subgroups of quaternion algebras over Q. In a family of such K3 surfaces, a surface has rho=20 if and only if it corresponds to a CM point on X. We use this to compute equations for Shimura curves, natural maps between them, and CM coordinates well beyond what could be done by working with the curves directly as we did in ``Shimura Curve Computations'' (1998) = Comment: 16 pages (1 figure drawn with the LaTeX picture environment); To appear in the proceedings of ANTS-VIII, Banff, May 200

Six line configurations and string dualities

We study the family of K3 surfaces of Picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines, and the family of its Van Geemen-Sarti partners, i.e., K3 surfaces with special Nikulin involutions, such that quotienting by the involution and blowing up recovers the former. We prove that the family of Van Geemen-Sarti partners is a four-parameter family of K3 surfaces with $H \oplus E_7(-1) \oplus E_7(-1)$ lattice polarization. We describe explicit Weierstrass models on both families using even modular forms on the bounded symmetric domain of type $IV$. We also show that our construction provides a geometric interpretation, called geometric two-isogeny, for the F-theory/heterotic string duality in eight dimensions. As a result, we obtain novel F-theory models, dual to non-geometric heterotic string compactifications in eight dimensions with two non-vanishing Wilson line parameters.Comment: 42 pages; minor typos corrected in version