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 Growth rate of ample heights and the Dynamical Mordell–Lang conjecture

2018 ◽  
Vol 14 (10) ◽  
pp. 2673-2685
Author(s):  
Kaoru Sano
Keyword(s):  
Growth Rate ◽  
Explicit Formula ◽  
Number Fields ◽  
Rational Points ◽  
Positive Answer ◽  
Projective Varieties ◽  
Smooth Projective Varieties

We provide an explicit formula on the growth rate of ample heights of rational points under iteration of endomorphisms of smooth projective varieties over number fields. As an application, we give a positive answer to a variant of the Dynamical Mordell–Lang conjecture for pairs of étale endomorphisms, which is also a variant of the original one stated by Bell, Ghioca, and Tucker in their monograph.

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.

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 Nonspecial varieties and generalised Lang–Vojta conjectures

2021 ◽  
Vol 9 ◽  
Author(s):  
Erwan Rousseau ◽  
Amos Turchet ◽  
Julie Tzu-Yueh Wang
Keyword(s):  
Number Field ◽  
Function Field ◽  
General Type ◽  
Function Fields ◽  
Rational Points ◽  
Special Variety ◽  
Recent Method ◽  
Dense Set

Abstract We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

scholarly journals Transcendental Liouville Inequalities on Projective Varieties

2019 ◽  
Vol 2020 (24) ◽  
pp. 9844-9886
Author(s):  
Carlo Gasbarri
Keyword(s):  
Line Bundle ◽  
Number Field ◽  
Projective Variety ◽  
Global Section ◽  
Rational Points ◽  
Liouville Type ◽  
Algebraic Point ◽  
Projective Varieties ◽  
Almost All

Abstract Let $p$ be an algebraic point of a projective variety $X$ defined over a number field. Liouville inequality tells us that the norm at $p$ of a non-vanishing integral global section of a hermitian line bundle over $X$ is zero or it cannot be too small with respect to the $\sup $ norm of the section itself. We study inequalities similar to Liouville’s for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture by Chudnovsky and give two applications to the growth of the number of rational points of bounded height on the image of an analytic map from a disk to a projective variety.

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.

ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)

2011 ◽  
Vol 21 (04) ◽  
pp. 595-614 ◽  
Author(s):  
S. LIRIANO ◽  
S. MAJEWICZ
Keyword(s):  
Algebraic Group ◽  
Free Group ◽  
Algebraic Variety ◽  
Commutator Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Geometric Invariants ◽  
Torus Knot ◽  
Irreducible Components ◽  
Irreducible Variety

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom (G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), …, N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim (RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, …, xn〉, and letG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, …, xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, …, xn〉, and A = PSL(2, ℂ), then Dim (RA(G)) = Max {3n, Dim (RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, …, xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)).

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