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Conjecture

Torsion Conjecture

Algebraic Geometry

In algebraic geometry and number theory, the torsion conjecture states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Statement
The order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field. 1
Proposed Year
1908 1
1908 is Beppo Levi's proposal for the elliptic-curve case, later proven by Barry Mazur (1977-1978); the conjecture for general abelian varieties remains open.
Progress Toward Resolution
Completely resolved in the case of elliptic curves: Barry Mazur proved it over the rationals (1977, 1978) and Loic Merel over any number field (1996). 1
Classification
Resolution Status
Partially Resolved 1
Prize Status
Prize Status (category)
No Prize Offered 1
Connections

In Branch

Number Theory, Branches of Mathematics
Source Torsion Conjecture (Wikipedia)

Posed By

Beppo Levi, Mathematicians
Source Torsion Conjecture (Wikipedia)
Sources
1. Torsion Conjecture (Wikipedia)
Wikimedia Foundation
  • Elliptic curves
    At the 1908 International Mathematical Congress in Rome, Levi conjectured that this is a complete list of torsion groups for elliptic curves over the rationals. The torsion conjecture for elliptic curves over the rationals was independently reformulated by Trygve Nagell (1952) and again by Andrew Ogg (1971), with the conjecture becoming commonly known as Ogg's conjecture.
  • Elliptic curves
    The torsion conjecture has been completely resolved in the case of elliptic curves.
  • Lead section, first sentence
    the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field
  • In Branch: Number Theory, Lead sentence
    In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian v
  • Posed By: Beppo Levi, History
    At the 1908 International Mathematical Congress in Rome, Levi conjectured that this is a complete list of torsion groups for elliptic curves over the rationals.
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