A372206 Largest prime that occurs infinitely often as an order of a rational point of an elliptic curve over a number field of degree n.

7, 13, 13, 17, 19, 19, 23, 23

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

Table of n, a(n) for n=1..8.

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

Cf. A372083 for the analog sequence of maximal primes that occur at least once.

Sequence in context: A135555 A243044 A229831 * A372083 A352444 A145009

Adjacent sequences: A372203 A372204 A372205 * A372207 A372208 A372210

Keyword

nonn,hard,more

Author

Thomas Preu, May 03 2024

Status

approved