Related papers: Quadratic residue patterns and point counting on K…

Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive $\ell$-tuples of quadratic residues, studying corresponding algebraic curves and their…

We report on our project to find explicit examples of $K3$ surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for…

We prove the unpolarized Shafarevich conjecture for K3 surfaces: the set of isomorphism classes of K3 surfaces over a fixed number field with good reduction away from a fixed and finite set of places is finite. Our proof is based on the…

We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry.

Generalizing the problem of counting rational points on curves and surfaces over finite fields, we consider the setting of $n \times n$ matrix points with finite field entries. We obtain exact formulas for matrix point counts on elliptic…

The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. Andr\'{e} proved this conjecture for…

This article reports on an approach to point counting on algebraic varieties over finite fields that is based on a detailed investigation of the $2$-adic orthogonal group. Combining the new approach with a $p$-adic method, we count the…

We prove that the moduli spaces of K3 surfaces with non-symplectic involutions are unirational. As a by-product we describe configuration spaces of 4<d<9 points in the projective plane as arithmetic quotients of type IV.

Using the theory of holes of the Leech lattice and Borcherds method for the computation of the automorphism group of a K3 surface, we give an effective bound for the set of isomorphism classes of projective models of fixed degree for…

We prove that for any of a wide class of elliptic surfaces $X$ defined over a number field $k$, if there is an algebraic point on $X$ that lies on only finitely many rational curves, then there is an algebraic point on $X$ that lies on no…

We study the distribution of algebraic points on K3 surfaces.

The name "K3 surfaces" was coined by A. Weil in 1957 when he formulated a research programme for these surfaces and their moduli. Since then, irreducible holomorphic symplectic manifolds have been introduced as a higher dimensional analogue…

We study surfaces constructed from groups of units in quaternion orders $\Lambda$ over the integers in real quadratic fields k. A short presentation of some general theory of such surfaces is given, in particular, we construct certain…

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously…

Let $D$ be an irreducible plane curve. In this article, we first introduce a notion of a quadratic residue curve mod $D$, and study quadratic residue concis $C$ mod an irreducible quartic curve $Q$. As an application, we study a dihedral…

K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we…

In general, not much is known about the arithmetic of K3 surfaces. Once the geometric Picard number, which is the rank of the Neron-Severi group over an algebraic closure of the base field, is high enough, more structure is known and more…

In this paper, we study the period mappings for the families of $K3$ surfaces derived from the $3$-dimensional $5$-verticed reflexive polytopes. We determine the lattice structures, the period differential equations and the projective…

We study the reduced descendent Gromov-Witten theory of K3 surfaces in primitive curve classes. We present a conjectural closed formula for the stationary theory, which generalizes the Bryan-Leung formula. We also prove a new recursion that…

We study the behavior of geometric Picard ranks of K3 surfaces over the rationals under reduction modulo primes. We compute these ranks for reductions of smooth quartic surfaces modulo all primes $p<2^{16}$ in several representative…