OFFSET
1,1
COMMENTS
Let F be the number field of degree n over which some relevant elliptic curve E is defined. Torsion subgroup refers to the torsion subgroup E(F)_{tors} of finite order F-rational points of E.
By a result of Pierre Parent building on the work of Merel Loïc, Barry Mazur, Andrew Ogg and others, the maximal p-power order of an F-rational point of E for any prime p is effectively bounded by P(n) = 129*(5^n-1)*(3n)^6. By a structure result behind the Mordell-Weil theorem -- E(F)_{tors} is isomorphic as an abelian group to Z/aZ x Z/bZ for some positive integers a, b -- the n-th term is effectively bounded by a(n) <= (P(n)^P(n))^2. While slightly better bounds exist, all bounds known as of 2024 are of a doubly exponential nature.
LINKS
Jennifer S. Balakrishnan, Barry Mazur and Netan Dogra, Ogg's Torsion conjecture: Fifty years later, arXiv:2307.04752 [math.NT], 2023.
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
In his survey article Andrew Sutherland lists all occurring subgroups up to degree 3 (the partially open case for degree 3 about "sporadic points" in Sutherland's survey has been resolved, cf. Balakrishnan et al.). The maximal order is realized by groups of the following isomorphism type: a(1)=12 by Z/12Z, a(2)=24 by Z/2Z x Z/12Z and a(3)=28 by Z/2Z x Z/14Z.
For n=4 all infinitely often occurring torsion subgroups are known, but the status of existence resp. isomorphism type of only finitely often occurring torsion subgroups is an open problem as of 2012.
CROSSREFS
A372549 is the analogous sequence of maximal infinitely often occurring orders. Both sequences agree up to n=3 and it is an open question whether this is a general phenomenon.
KEYWORD
nonn,hard,more,bref
AUTHOR
Thomas Preu, May 05 2024
STATUS
approved