scholarly journals Lang’s conjecture and sharp height estimates for the elliptic curves $y^{2}=x^{3}+b$

2016 ◽  
pp. 1-28
Author(s):  
Paul Voutier ◽  
Minoru Yabuta
Keyword(s):  
Elliptic Curves ◽  
Height Estimates ◽  
Lang's Conjecture ◽  
Lang’S Conjecture

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.

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.

scholarly journals The Erdős–Ulam problem, Lang's conjecture and uniformity

2020 ◽  
Vol 52 (6) ◽  
pp. 1053-1063 ◽  
Author(s):  
Kenneth Ascher ◽  
Lucas Braune ◽  
Amos Turchet
Keyword(s):  
Lang's Conjecture ◽  
Lang’S Conjecture

Going-Down Results for Ci-Fields

2006 ◽  
Vol 49 (1) ◽  
pp. 11-20
Author(s):  
Anthony J. Bevelacqua ◽  
Mark J. Motley
Keyword(s):  
Function Fields ◽  
Algebraic Extension ◽  
Real Closed Field ◽  
Closed Field ◽  
Difficult Case ◽  
Lang's Conjecture ◽  
Lang’S Conjecture

AbstractWe search for theorems that, given a Ci-field K and a subfield k of K, allow us to conclude that k is a Cj -field for some j. We give appropriate theorems in the case K = k(t) and K = k((t)). We then consider the more difficult case where K/k is an algebraic extension. Here we are able to prove some results, and make conjectures. We also point out the connection between these questions and Lang's conjecture on nonreal function fields over a real closed field.

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