scholarly journals Remarks on endomorphisms and rational points

2011 ◽  
Vol 147 (6) ◽  
pp. 1819-1842 ◽  
Author(s):  
E. Amerik ◽  
F. Bogomolov ◽  
M. Rovinsky
Keyword(s):  
Fixed Point ◽  
Number Field ◽  
Algebraic Variety ◽  
Rational Points ◽  
Potential Density ◽  
Cubic Fourfold

AbstractLet X be an algebraic variety and let f:X−−→X be a rational self-map with a fixed point q, where everything is defined over a number field K. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.

scholarly journals Higher-dimensional Calabi–Yau varieties with dense sets of rational points

2021 ◽  
Author(s):  
Fumiaki Suzuki
Keyword(s):  
Number Field ◽  
Rational Points ◽  
Enriques Surface ◽  
Potential Density ◽  
Higher Dimensional ◽  
Dense Sets ◽  
Arbitrary Dimensions

AbstractWe construct higher-dimensional Calabi–Yau varieties defined over a given number field with Zariski dense sets of rational points. We give two elementary constructions in arbitrary dimensions as well as another construction in dimension three which involves certain Calabi–Yau threefolds containing an Enriques surface. The constructions also show that potential density holds for (sufficiently) general members of the families.

scholarly journals ABELIAN FIBRATIONS AND RATIONAL POINTS ON SYMMETRIC PRODUCTS

2000 ◽  
Vol 11 (09) ◽  
pp. 1163-1176 ◽  
Author(s):  
BRENDAN HASSETT ◽  
YURI TSCHINKEL
Keyword(s):  
Number Field ◽  
Finite Extension ◽  
Rational Points ◽  
Symmetric Product ◽  
Symmetric Power ◽  
K3 Surface ◽  
Potential Density ◽  
Symmetric Products ◽  
Geometric Problem ◽  
Basic Construction

Given a variety over a number field, are its rational points potentially dense, i.e. does there exist a finite extension over which rational points are Zariski dense? We study the question of potential density for symmetric products of surfaces. Contrary to the situation for curves, rational points are not necessarily potentially dense on a sufficiently high symmetric product. Our main result is that rational points are potentially dense for the Nth symmetric product of a K3 surface, where N is explicitly determined by the geometry of the surface. The basic construction is that for some N, the Nth symmetric power of a K3 surface is birational to an Abelian fibration over ℙN. It is an interesting geometric problem to find the smallest N with this property.

scholarly journals On the density of rational points on elliptic fibrations

1999 ◽  
Vol 1999 (511) ◽  
pp. 87-93 ◽  
Author(s):  
F. A Bogomolov ◽  
Yu Tschinkel
Keyword(s):  
Number Field ◽  
Algebraic Variety ◽  
General Type ◽  
Finite Extension ◽  
Rational Points ◽  
Projective Varieties ◽  
Elliptic Fibrations ◽  
Geometric Invariants ◽  
Smooth Projective Varieties

1. Introduction Let X be an algebraic variety defined over a number field F. We will say that rational points are potentially dense if there exists a finite extension K/F such that the set of K-rational points X(K) is Zariski dense in X. The main problem is to relate this property to geometric invariants of X. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]).

scholarly journals A Gap Principle for Subvarieties with Finitely Many Periodic Points

2020 ◽  
Vol 63 (2) ◽  
pp. 382-392
Author(s):  
Keping Huang
Keyword(s):  
Number Field ◽  
Algebraic Variety ◽  
Initial Point ◽  
Periodic Points ◽  
Gap Principle ◽  
Special Case

AbstractLet $f:X\rightarrow X$ be a quasi-finite endomorphism of an algebraic variety $X$ defined over a number field $K$ and fix an initial point $a\in X$. We consider a special case of the Dynamical Mordell–Lang Conjecture, where the subvariety $V$ contains only finitely many periodic points and does not contain any positive-dimensional periodic subvariety. We show that the set $\{n\in \mathbb{Z}_{{\geqslant}0}\mid f^{n}(a)\in V\}$ satisfies a strong gap principle.

scholarly journals A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

2004 ◽  
Vol 47 (3) ◽  
pp. 398-406
Author(s):  
David McKinnon
Keyword(s):  
Number Field ◽  
Open Subset ◽  
Rational Points ◽  
Rational Curves ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Geometric Genus ◽  
Finite Set ◽  
Genus 2 ◽  
Kummer Surfaces

AbstractLet V be a K3 surface defined over a number field k. The Batyrev-Manin conjecture for V states that for every nonempty open subset U of V, there exists a finite set ZU of accumulating rational curves such that the density of rational points on U − ZU is strictly less than the density of rational points on ZU. Thus, the set of rational points of V conjecturally admits a stratification corresponding to the sets ZU for successively smaller sets U.In this paper, in the case that V is a Kummer surface, we prove that the Batyrev-Manin conjecture for V can be reduced to the Batyrev-Manin conjecture for V modulo the endomorphisms of V induced by multiplication by m on the associated abelian surface A. As an application, we use this to show that given some restrictions on A, the set of rational points of V which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Batyrev-Manin type.

scholarly journals Trinomials defining quintic number fields

2017 ◽  
Vol 13 (07) ◽  
pp. 1881-1894 ◽  
Author(s):  
Jesse Patsolic ◽  
Jeremy Rouse
Keyword(s):  
Elliptic Curve ◽  
Number Field ◽  
Number Fields ◽  
Rational Points

Given a quintic number field K/ℚ, we study the set of irreducible trinomials, polynomials of the form x5 + ax + b, that have a root in K. We show that there is a genus 4 curve CK whose rational points are in bijection with such trinomials. This curve CK maps to an elliptic curve defined over a number field, and using this map, we are able (in some cases) to determine all the rational points on CK using elliptic curve Chabauty.

scholarly journals THE MULTILINEAR SUPPORT PROBLEM FOR PRODUCTS OF ABELIAN VARIETIES AND TORI

2012 ◽  
Vol 08 (01) ◽  
pp. 255-264
Author(s):  
ANTONELLA PERUCCA
Keyword(s):  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Rational Points ◽  
Dependency Relation

Let G be the product of an abelian variety and a torus defined over a number field K. The aim of this paper is detecting the dependence among some given rational points of G by studying their reductions modulo all primes of K. We show that if some simple conditions on the order of the reductions of the points are satisfied then there must be a dependency relation over the ring of K-endomorphisms of G. We generalize Larsen's result on the support problem to several points on products of abelian varieties and tori.

scholarly journals Elliptic Fibrations and the Hilbert Property

2019 ◽  
Author(s):  
Julian Lawrence Demeio
Keyword(s):  
Number Field ◽  
Algebraic Variety ◽  
Rational Point ◽  
K3 Surfaces ◽  
Sufficient Condition ◽  
Algebraic Varieties ◽  
Elliptic Fibrations ◽  
Kummer Surfaces ◽  
Cubic Hypersurfaces

Abstract For a number field $K$, an algebraic variety $X/K$ is said to have the Hilbert Property if $X(K)$ is not thin. We are going to describe some examples of algebraic varieties, for which the Hilbert Property is a new result. The first class of examples is that of smooth cubic hypersurfaces with a $K$-rational point in ${\mathbb{P}}_n/K$, for $n \geq 3$. These fall in the class of unirational varieties, for which the Hilbert Property was conjectured by Colliot-Thélène and Sansuc. We then provide a sufficient condition for which a surface endowed with multiple elliptic fibrations has the Hilbert Property. As an application, we prove the Hilbert Property of a class of K3 surfaces, and some Kummer surfaces.

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