OFFSET
1,1
COMMENTS
A rational point means an F-rational point where F is a number field of degree n over which some relevant elliptic curve E is defined.
By a result of Pierre Parent building on the work of Merel Loïc, Barry Mazur, Andrew Ogg, and others the n-th term is effectively bounded by 65*(3^n-1)*(2n)^6. Derickx et al. (pp. 21ff) give the slightly better bound (3^(n/2)+1)^2 for n >= 3 from Oesterlé (1994).
Denoting the set of all primes <= m by Primes(m) it holds that the set of all primes realized as the order of a rational point of an elliptic curve for a degree n number field is Primes(a(n)) for 1 <= n <= 8 except for n = 6 where it is Primes(19) union {37}.
Derickx et al. (p. 8) comment on the difficulties of finding the next numbers in that sequence.
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.
Maleeha Khawaja, Torsion primes for elliptic curves over degree 8 number fields, arXiv:2304.14284 [math.NT], 2023-2024.
Pierre Parent, Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres, arXiv:1307.5719 [math.NT], 2013-2014.
Andrew V. Sutherland, Torsion subgroups of elliptic curves over number fields.
EXAMPLE
Points of finite order are collected in the torsion subgroup T = E(F)_{tors} of the F-rational points of an elliptic curve E. For n <= 4 the prime a(n) is realized by a cyclic group C_{a(n)} of order a(n) according to the survey article of Sutherland, explicitly a(1)=7 by T=C_7, a(2)=13 by T=C_13, a(3)=13 by T=C_13, a(4)=17 by T=C_17.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Thomas Preu, May 03 2024
EXTENSIONS
a(8) based on the paper of Maleeha Khawaja added by Thomas Preu, Aug 01 2024
STATUS
approved