In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for
abelian varieties states that the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
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. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field. The torsion conjecture has been completely resolved in the case of
elliptic curves
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
.
Elliptic curves
From 1906 to 1911,
Beppo Levi published a series of papers investigating the possible finite orders of points on elliptic curves over the rationals. He showed that there are infinitely many elliptic curves over the rationals with the following torsion groups:
* ''C''
''n'' with 1 ≤ ''n'' ≤ 10, where ''C''
''n'' denotes the
cyclic group of order ''n'';
* ''C''
12;
* ''C''
2n × ''C''
2 with 1 ≤ ''n'' ≤ 4, where × denotes the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
.
At the 1908
International Mathematical Congress
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be rena ...
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 and again by , with the conjecture becoming commonly known as Ogg's conjecture.
drew the connection between the torsion conjecture for elliptic curves over the rationals and the theory of
classical modular curve In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation
:,
such that is a point on the curve. Here denotes the -invariant.
The curve is sometimes called , though often that notation is used f ...
s. In the early 1970s, the work of Gérard Ligozat,
Daniel Kubert
Daniel Sion Kubert (; October 18, 1947 – January 5, 2010) was an American mathematician who introduced modular unit In mathematics, modular units are certain units of rings of integers of fields of modular functions, introduced by . They are fun ...
,
Barry Mazur
Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in ...
, and
John Tate showed that several small values of ''n'' do not occur as orders of torsion points on elliptic curves over the rationals. proved the full torsion conjecture for elliptic curves over the rationals. His techniques were generalized by and , who obtained uniform boundedness for
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...
s and number fields of degree at most 8 respectively. Finally, proved the conjecture for elliptic curves over any number field.
An effective bound for the size of the torsion group in terms of the degree of the number field was given by . A complete list of possible torsion groups has also been given for elliptic curves over quadratic number fields. There are substantial partial results for quartic and quintic number fields .
See also
*
Bombieri–Lang conjecture In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type.
Statement
The weak Bombie ...
*
Uniform boundedness conjecture for preperiodic points Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics ...
*
Uniform boundedness conjecture for rational points
In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field K and a positive integer g \geq 2 that there exists a number N(K,g) depending only on K and g such that for any algebraic curve C ...
References
Bibliography
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Abelian varieties
Conjectures
Diophantine geometry
Theorems in number theory
Theorems in algebraic geometry
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