Neighbors and arithmetic of isogenous K3 surfaces

TL;DR

This paper employs lattice theory to analyze the isogeny classes of K3 surfaces, constructing isogenies via neighboring lattices, and explores their fields of definition, relating to conjectures on Brauer groups and Néron-Severi lattices.

Contribution

It introduces a lattice-theoretic approach to study isogenies of K3 surfaces, including construction methods and field of definition analysis, linking to conjectures in algebraic geometry.

Findings

Constructed isogenies using Kneser neighboring lattice method.

Determined fields of definition for isogenous K3 surfaces.

Connected results to conjectures on Brauer groups and Néron-Severi lattices.

Abstract

We use lattice theory to study the isogeny class of a K3 surface. Starting from isotropic Brauer classes, we construct isogenies via Kneser method of neighboring lattices. We also determine the fields of definition of isogenous K3 surfaces, and study Kneser construction over number fields. We then apply our results to relate conjectures about the finiteness of Brauer groups and N\'{e}ron-Severi lattices of K3 surfaces.