Rational points and derived equivalence

Abstract

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over $\mathbb {Q}$ and $\mathbb {F}_q(t)$ , and conclude with a pair of hyperkähler 4-folds over $\mathbb {Q}$ . The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.

Derived equivalences of gerbey curves

“…Proof. First we argue that it is enough to prove the claim when k is algebraically closed, again mimicking the argument given in [1]. If…”

“…We will set up a theory of integral transforms for derived categories of G m -gerbes over proper schemes and prove that many facts that hold for integral transforms of schemes also hold for integral transforms of such stacks. 1…”

A.T.V.’s for (Geometric) Off-roading: A Gentle Introduction to Arithmetic Toric Varieties

“…The use of 𝖣 𝖻 (𝑋) in determining whether a variety admits rational points was motivated by a question of H. Esnault. This was answered in the negative in [AAFH19]. Related work was carried out in [AKW17], showing that twisted forms cannot always be disinguished by the derived category, even in nice situations in low dimension.…”

Equivariant derived equivalence and rational points on K3 surfaces

“…Related questions were considered by [AAHF21] (hyperkähler fourfolds) and twisted K3 surfaces [ADPZ17]; here the existence of rational points is not compatible with derived equivalence. The case of torsors for abelian varieties is addressed in [AKW17].…”