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 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.

scholarly journals Bounds on $2$-torsion in class groups of number fields and integral points on elliptic curves

2020 ◽  
Vol 33 (4) ◽  
pp. 1087-1099 ◽  
Author(s):  
M. Bhargava ◽  
A. Shankar ◽  
T. Taniguchi ◽  
F. Thorne ◽  
J. Tsimerman ◽  
...  
Keyword(s):  
Elliptic Curves ◽  
Number Fields ◽  
Class Groups ◽  
Integral Points
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