On rational points on projective varieties defined over a complete valuation field

1966 ◽  
pp. 84-89
Author(s):  
Tsuneo Tamagawa
Keyword(s):  
Rational Points ◽  
Projective Varieties ◽  
Valuation Field

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.

A SIMPLE PROOF OF CHEBOTAREV’S DENSITY THEOREM OVER FINITE FIELDS

2018 ◽  
Vol 98 (2) ◽  
pp. 196-202
Author(s):  
STEVE MEAGHER
Keyword(s):  
Conjugacy Class ◽  
Finite Fields ◽  
Simple Proof ◽  
Function Fields ◽  
Rational Points ◽  
Density Theorem ◽  
Chebotarev Density Theorem ◽  
Projective Varieties

We present a simple proof of the Chebotarev density theorem for finite morphisms of quasi-projective varieties over finite fields following an idea of Fried and Kosters for function fields. The key idea is to interpret the number of rational points with a given Frobenius conjugacy class as the number of rational points of a twisted variety, which is then bounded by the Lang–Weil estimates.

A new method of solving certain quartic and higher degree diophantine equations

2018 ◽  
Vol 14 (08) ◽  
pp. 2129-2154
Author(s):  
Ajai Choudhry
Keyword(s):  
Homogeneous Polynomial ◽  
Diophantine Equations ◽  
Rational Points ◽  
Simultaneous Equations ◽  
New Method ◽  
Rational Solutions ◽  
Projective Varieties ◽  
Integer Solutions ◽  
High Degree

In this paper, we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be applied to some diophantine systems in five or more variables. Under certain conditions, the method yields an arbitrarily large number of integer solutions of such diophantine equations and diophantine systems, two examples being a sextic equation in four variables and two simultaneous equations of degrees four and six in six variables. We also simultaneously obtain arbitrarily many rational solutions of certain related nonhomogeneous equations of high degree. We obtain these solutions without finding a curve of genus 0 or 1 on the variety defined by the equations concerned. It appears that there exist projective varieties on which there are an arbitrarily large number of rational points and which do not contain a curve of genus 0 or 1 with infinitely many rational points.

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 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]).

Cohomological Conditions on Endomorphisms of Projective Varieties

2017 ◽  
Vol 145 (3) ◽  
pp. 449-468 ◽  
Author(s):  
Holly Krieger ◽  
Paul Reschke
Keyword(s):  
Projective Varieties
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