The difference between the Weil height and the canonical height on elliptic curves

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

International Journal of Number Theory ◽

10.1142/s1793042113500176 ◽

2013 ◽

Vol 09 (05) ◽

pp. 1141-1170 ◽

Cited By ~ 1

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.

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Points of low height on elliptic surfaces with torsion

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157009000151 ◽

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.

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Modular invariants and isogenies

International Journal of Number Theory ◽

10.1142/s1793042119500295 ◽

2019 ◽

Vol 15 (03) ◽

pp. 569-584

Author(s):

Fabien Pazuki

Keyword(s):

Elliptic Curves ◽

Abelian Varieties ◽

Classical Estimate ◽

Modular Invariants ◽

Explicit Bounds ◽

The Difference ◽

Faltings Height ◽

Modular Polynomials

We provide explicit bounds on the difference of heights of the [Formula: see text]-invariants of isogenous elliptic curves defined over [Formula: see text]. The first one is reminiscent of a classical estimate for the Faltings height of isogenous abelian varieties, which is indeed used in the proof. We also use an explicit version of Silverman’s inequality and isogeny estimates by Gaudron and Rémond. We give applications in the study of Vélu’s formulas and of modular polynomials.

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