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 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 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 Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

2021 ◽  
Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Waring’S Problem ◽  
Waring Problem ◽  
Waring's Problem ◽  
Main Conjecture ◽  
Higher Dimensional ◽  
Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

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.

ON SEXTIC TRINOMIALS WITH GALOIS GROUP C6, S3 OR C3 × S3

2012 ◽  
Vol 12 (01) ◽  
pp. 1250128 ◽  
Author(s):  
STEPHEN C. BROWN ◽  
BLAIR K. SPEARMAN ◽  
QIDUAN YANG
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Rational Points ◽  
Genus 2

We characterize irreducible trinomials x6 + Ax + B with coefficients in a number field K which have Galois group C6, S3 or C3 × S3. This characterization relates these trinomials to the K-rational points on a genus 2 curve. We determine these trinomials explicitly in the case K = ℚ.

scholarly journals Arithmetic hyperbolicity: automorphisms and persistence

2021 ◽  
Author(s):  
Ariyan Javanpeykar
Keyword(s):  
Automorphism Group ◽  
Number Field ◽  
Projective Variety ◽  
Rational Points ◽  
General Criterion ◽  
Finitely Generated ◽  
Integral Points ◽  
Lang's Conjecture ◽  
Lang’S Conjecture

AbstractWe show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang’s conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many rational points has only finitely many automorphisms. Moreover, we investigate to what extent finiteness of S-integral points on a variety over a number field persists over finitely generated fields. To this end, we introduce the class of mildly bounded varieties and prove a general criterion for proving this persistence.

scholarly journals Problèmes D'effectivité sur les Quartiques de Fermat

2007 ◽  
Vol 75 (1) ◽  
pp. 91-119
Author(s):  
Élie Cali ◽  
Alain Kraus
Keyword(s):  
Number Field ◽  
Rational Points

Let K be a number field. An element b ∈ K* being given, let Cb be the curve defined over K by the equation x4 + y4 = bz4. Let Cb(K) be the set of the K-rational points of Cb. This paper uses Dem'janenko and Manin type methods to obtain effective criteria for Cb(K) to be empty.

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