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.

Reductions of one-dimensional tori

2016 ◽  
Vol 13 (06) ◽  
pp. 1473-1489 ◽  
Author(s):  
Antonella Perucca
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Rational Points ◽  
Splitting Field ◽  
Dimensional Torus ◽  
Finitely Generated ◽  
One Dimensional ◽  
Finitely Generated Group ◽  
Dirichlet Density ◽  
Number Formula

Consider a non-split one-dimensional torus defined over a number field [Formula: see text]. For a finitely generated group [Formula: see text] of rational points and for a prime number [Formula: see text], we investigate for how many primes [Formula: see text] of [Formula: see text] the size of the reduction of [Formula: see text] modulo [Formula: see text] is coprime to [Formula: see text]. We provide closed formulas for the corresponding Dirichlet density in terms of finitely many computable parameters. To achieve this, we determine in general which torsion fields and Kummer extensions contain the splitting field.

scholarly journals LANG'S CONJECTURE AND SHARP HEIGHT ESTIMATES FOR THE ELLIPTIC CURVES y2 = x3 + ax

2013 ◽  
Vol 09 (05) ◽  
pp. 1141-1170 ◽  
Author(s):  
PAUL VOUTIER ◽  
MINORU YABUTA
Keyword(s):  
Lower Bound ◽  
Elliptic Curves ◽  
Lower Bounds ◽  
Rational Points ◽  
Upper And Lower Bounds ◽  
Canonical Height ◽  
The Difference ◽  
Lang's Conjecture ◽  
Lang’S Conjecture ◽  
Logarithmic Height

For elliptic curves given by the equation Ea : y2 = x3 + ax, we establish the best-possible version of Lang's conjecture on the lower bound for the canonical height of non-torsion rational points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.

Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces

2014 ◽  
Vol 2014 (697) ◽  
pp. 1-14 ◽  
Author(s):  
Sai-Kee Yeung
Keyword(s):  
Number Field ◽  
Rational Points ◽  
Two Dimensional ◽  
Hyperbolic Surfaces ◽  
Integral Points ◽  
Negatively Curved ◽  
Ball Quotients ◽  
Complex Ball ◽  
Compact Surfaces ◽  
Dimensional Ball

AbstractThe first goal of this paper is to study the question of finiteness of integral points on a cofinite non-compact complex two-dimensional ball quotient defined over a number field. Along the process we will also consider some negatively curved compact surfaces. Using some fundamental results of Faltings, the question is to reduce to a conjecture of Borel about existence of virtual holomorphic one-forms on cofinite non-cocompact complex ball quotients. The second goal of this paper is to study the conjecture on such non-compact surfaces.

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 Elliptic Curves over the Perfect Closure of a Function Field

2010 ◽  
Vol 53 (1) ◽  
pp. 87-94
Author(s):  
Dragos Ghioca
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Positive Characteristic ◽  
Rational Points ◽  
Finitely Generated

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

Schmidt’s subspace theorem for non-subdegenerate families of hyperplanes

2021 ◽  
pp. 1-18
Author(s):  
Duc Hiep Pham
Keyword(s):  
Type Theorem ◽  
Exceptional Sets ◽  
Subspace Theorem ◽  
Lang's Conjecture ◽  
Lang’S Conjecture

In this paper, we establish a Schmidt’s subspace theorem for non-subdegenerate families of hyperplanes. In particular, our result improves the previous result on Schmidt’s subspace type theorem for the case of non-degenerate families of hyperplanes, and furthermore, also shows the sharpness of the condition of non-subdegeneracy. As a consequence, we deduce a version of Lang’s conjecture on exceptional sets in the case of complements of hyperplanes.

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