OFFSET
1,1
COMMENTS
Rational point is understood as F-rational point for F the number field of degree n over which some relevant elliptic curve E is defined.
By a result of Pierre Parent building on work of Merel Loïc, Barry Mazur, Andrew Ogg and others the n-th term is effectively bounded by 65*(3^n-1)*(2n)^6.
Denoting the set of all primes at most m by Primes(m) it holds that the set of all primes realized infinitely often as order of a rational point of an elliptic curve for a degree n number field is Primes(a(n)) for 1<=n<=8. It is unclear if this pattern continues.
According to Derickx, Kamienny et al. (p. 2) the computation of the numbers in this sequence is a consequence of computing gonalities of certain modular curves. Derickx and van Hoeij describe in principle an algorithm that computes such gonalities, but in Remark 4 (p. 14) they comment on the difficulty of computing those gonalities for n>=9.
LINKS
Jennifer S. Balakrishnan, Barry Mazur and Netan Dogra, Ogg's Torsion conjecture: Fifty years later, arXiv:2307.04752 [math.NT], 2023.
Maarten Derickx, Sheldon Kamienny, William Stein and Michael Stoll, Torsion points on elliptic curves over number fields of small degree, arXiv:1707.00364 [math.NT], 2017-2021.
Maarten Derickx and Mark van Hoeij, Gonality of the modular curve X1(N), arXiv:1307.5719 [math.NT], 2013-2014.
Pierre Parent, Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres, arXiv:alg-geom/9611022, 1996.
Andrew V. Sutherland, Torsion subgroups of elliptic curves over number fields.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Thomas Preu, May 03 2024
STATUS
approved