We study K3 surfaces over non-closed fields and relate the notion of derived
equivalence to arithmetic problems.Comment: 30 page

Hassett, Brendan

Tschinkel, Yuri

English

arXiv

The geometry of vector bundles and derived categories on complex K3 surfaces has developed rapidly since Mukai&apos;s seminal work [Muk87]. Many foundational questions have been answered: • the existence of vector bundles and twisted sheaves with pre-scribed invariants; • geometric interpretations of isogenies between K3 surfaces [Orl97, C l00]; • the global Torelli theorem for holomorphic symplectic manifolds [Ver13, Huy12b]; • the analysis of stability conditions and its implications for bi-rational geometry of moduli spaces of vector bundles and more general objects in the derived category [BMT14, BM14, Bri07]. Given the precision and power of these results, it is natural to seek arithmetic applications of this circle of ideas. Questions about zero cycles on K3 surfaces have attracted the attention of Beauville-Voisin [BV04], Huybrechts [Huy12a], and other authors. Our focus in this note is on rational points over non-closed fields of arithmetic interest. We seek to relate the notion of derived equivalence to arithmetic problems over various fields. Our guiding questions are: Question 1. Let X and Y be K3 surfaces, derived equivalent over a field F. Does the existence/density of rational points of X imply the same for Y? Given α ∈ Br(X), let (X,α) denote the twisted K3 surface associated with α, i.e., if P → X is an étale projective bundle representing α, of relative dimension r − 1 then (X,α)  = [P / SLr]. Question 2. Suppose that (X,α) and (Y, β) are derived equivalent over F. Does the existence of a rational point on the former imply the same for the latter

Rational points on K3 surfaces and derived equivalence

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