Ronald van Luijk

On character varieties of two-bridge knot groups

We find explicit models for the PSL2(C)and SL2(C)-character varieties of the fundamental groups of complements in S of an infinite family of two-bridge knots that contains the twist knots. We compute the genus of the components of these character varieties, and deduce upper bounds on the degree of the associated trace fields. We also show that these knot complements are fibered if and only if they are commensurable to a fibered knot complement in a Z/2Z-homology sphere, resolving a conjecture of Hoste and Shanahan.

Nontrivial elements of Sha explained through K3 surfaces

We present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.

Computing Néron–Severi groups and cycle class groups

Assuming the Tate conjecture and the computability of etale cohomology with finite coefficients, we give an algorithm that computes the Neron-Severi group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equivalence classes of codimension p cycles for any p.

K3 surfaces with Picard number three and canonical vector heights

In this paper we construct the first known explicit family of K3 surfaces defined over the rationals that are proved to have geometric Picard number 3. This family is dense in one of the components of the moduli space of all polarized K3 surfaces with Picard number at least 3. We also use an example from this family to fill a gap in an earlier paper by the first author. In that paper, an argument for the nonexistence of canonical vector heights on K3 surfaces of Picard number 3 was given, based on an explicit surface that was not proved to have Picard number 3. We redo the computations for one of our surfaces and come to the same conclusion.

Unirationality of del Pezzo surfaces of degree 2 over finite fields

We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and Va rilly-Alvarado, who had proved this for all but three surfaces. Over general fields of characteristic not equal to two, we state sufficient conditions for a del Pezzo surface of degree two to be unirational.

Two-coverings of Jacobians of curves of genus 2

Given a curve C of genus 2 defined over a field k of characteristic different from 2, with a Jacobian variety J, we show that the two-coverings corresponding to elements of a large subgroup of H1(Gal(ks/k), J[2](ks)) (containing the Selmer group when k is a global field) can be embedded as an intersection of 72 quadrics in ℙ15k, just as the Jacobian J itself. Moreover, we actually give explicit equations for the models of these twists in the generic case, extending the work of Gordon and Grant which applied only to the case when all Weierstrass points are rational. In addition, we describe elegant equations of the Jacobian itself, and answer a question of Cassels and Flynn concerning a map from the Kummer surface in ℙ3 to the desingularized Kummer surface in ℙ5.

Non-Euclidean pythagorean triples, a problem of euler, and rational points on K3 surfaces

Author(s): Hartshorne, Robin; Luijk, Ronald van | Abstract: We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Eulers work on the second problem with a modern point of view.

The diameter of the circumcircle of a Heron triangle

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