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A372549
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Largest infinitely often occurring order of a torsion subgroup of an elliptic curve over a number field of degree n.
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1
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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 types: a(1)=12 by Z/12Z, a(2)=24 by Z/2Z x Z/12Z, a(3)=28 by Z/2Z x Z/14Z and a(4)=36 by Z/2Z x Z/18Z as well as Z/6Z x Z/6Z.
Over degree n=3 fields the torsion group Z/21Z arises exactly once and in particular not infinitely often. This does not interfere with the statistics of this sequence.
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CROSSREFS
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A372548 is the analogous sequence of maximal (potentially only finitely often) occurring orders. Both sequences agree up to n=3 and it is an open question whether this is a general phenomenon.
Related are A372083 and A372206, which are about maximal (infinitely often occurring) prime divisors of such torsion groups.
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KEYWORD
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nonn,hard,more,bref
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AUTHOR
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STATUS
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approved
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