scholarly journals A statistical view on the conjecture of Lang about the canonical height on elliptic curves

2019 ◽  
Vol 372 (12) ◽  
pp. 8347-8361
Author(s):  
Pierre Le Boudec
Keyword(s):  
Elliptic Curves ◽  
Canonical Height

scholarly journals Points of low height on elliptic surfaces with torsion

2010 ◽  
Vol 13 ◽  
pp. 370-387
Author(s):  
Sonal Jain
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Open Subset ◽  
Function Field ◽  
Rational Point ◽  
Torsion Point ◽  
Integral Points ◽  
Elliptic Surfaces ◽  
Canonical Height

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.

The canonical height and integral points on elliptic curves

1988 ◽  
Vol 93 (2) ◽  
pp. 419-450 ◽  
Author(s):  
M. Hindry ◽  
J. H. Silverman
Keyword(s):  
Elliptic Curves ◽  
Integral Points ◽  
Canonical Height

scholarly journals Computing canonical heights on elliptic curves in quasi-linear time

2016 ◽  
Vol 19 (A) ◽  
pp. 391-405 ◽  
Author(s):  
J. Steffen Müller ◽  
Michael Stoll
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Linear Time ◽  
Integer Factorization ◽  
Canonical Height ◽  
The Difference

We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the naive height into an archimedean and a non-archimedean term. Our main contribution is an algorithm for the computation of the non-archimedean term that requires no integer factorization and runs in quasi-linear time.

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