scholarly journals Rational Points on Twisted K3 Surfaces and Derived Equivalences

2017 ◽  
pp. 13-28 ◽  
Author(s):  
Kenneth Ascher ◽  
Krishna Dasaratha ◽  
Alexander Perry ◽  
Rong Zhou
Keyword(s):  
K3 Surfaces ◽  
Rational Points ◽  
Derived Equivalences

scholarly journals Autoequivalences of twisted K3 surfaces

2019 ◽  
Vol 155 (5) ◽  
pp. 912-937 ◽  
Author(s):  
Emanuel Reinecke
Keyword(s):  
K3 Surfaces ◽  
K3 Surface ◽  
Lattice Structures ◽  
Hodge Structures ◽  
Derived Equivalences

Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index $1$or $2$in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.

scholarly journals Derived equivalences of K3 surfaces and orientation

2009 ◽  
Vol 149 (3) ◽  
pp. 461-507 ◽  
Author(s):  
Daniel Huybrechts ◽  
Emanuele Macrì ◽  
Paolo Stellari
Keyword(s):  
K3 Surfaces ◽  
Derived Equivalences

scholarly journals K3 surfaces, rational curves, and rational points

2010 ◽  
Vol 130 (7) ◽  
pp. 1470-1479 ◽  
Author(s):  
Arthur Baragar ◽  
David McKinnon
Keyword(s):  
K3 Surfaces ◽  
Rational Points ◽  
Rational Curves

scholarly journals Density of rational points on elliptic K3 surfaces

2000 ◽  
Vol 4 (2) ◽  
pp. 351-368 ◽  
Author(s):  
F. A. Bogomolov ◽  
Yu. Tschinkel
Keyword(s):  
K3 Surfaces ◽  
Rational Points

Rational points on K3 surfaces in ?1 � ?1 � ?1

1996 ◽  
Vol 305 (1) ◽  
pp. 541-558 ◽  
Author(s):  
A. Baragar
Keyword(s):  
K3 Surfaces ◽  
Rational Points

scholarly journals Rational points and derived equivalence

2021 ◽  
Vol 157 (5) ◽  
pp. 1036-1050
Author(s):  
Nicolas Addington ◽  
Benjamin Antieau ◽  
Katrina Honigs ◽  
Sarah Frei
Keyword(s):  
Rational Point ◽  
Source Code ◽  
Rational Points ◽  
The Other ◽  
Online Version ◽  
Hasse Principle ◽  
Derived Equivalence ◽  
Derived Equivalences ◽  
Closed Fields ◽  
Supplementary Material

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.

scholarly journals Rational points on elliptic K3 surfaces of quadratic twist type

2020 ◽  
Author(s):  
Zhizhong Huang
Keyword(s):  
Elliptic Curves ◽  
K3 Surfaces ◽  
Rational Points ◽  
Zariski Topology ◽  
Quadratic Twists ◽  
Quadratic Twist ◽  
Quartic Polynomials

Abstract In studying rational points on elliptic K3 surfaces of the form $$\begin{equation*} f(t)y^2=g(x), \end{equation*}$$ where f, g are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell–Weil rank, and we relate it to the Hilbert property. Applying to surfaces of Cassels–Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.

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