%I #16 May 27 2024 23:11:52

%S 12,24,28

%N Largest order of a torsion subgroup of an elliptic curve over a number field of degree n.

%C 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.

%C 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.

%H Jennifer S. Balakrishnan, Barry Mazur and Netan Dogra, <a href="https://arxiv.org/abs/2307.04752">Ogg's Torsion conjecture: Fifty years later</a>, arXiv:2307.04752 [math.NT], 2023.

%H Pierre Parent, <a href="https://arxiv.org/abs/alg-geom/9611022">Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres</a>, arXiv:1307.5719 [math.NT], 2013-2014.

%H Andrew V. Sutherland, <a href="https://math.mit.edu/~drew/MazursTheoremSubsequentResults.pdf">Torsion subgroups of elliptic curves over number fields</a>.

%e 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.

%e 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.

%Y 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.

%Y Related are A372083 and A372206, which are about maximal (infinitely often occurring) prime divisors of such torsion groups.

%K nonn,hard,more,bref

%O 1,1

%A _Thomas Preu_, May 05 2024