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
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, the torsion conjecture or uniform boundedness conjecture for torsion points for
abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular function ...
states that the
order of the
torsion group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements.
For examp ...
of an abelian variety over a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
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
Beppo Levi (14 May 1875 – 28 August 1961) was an Italian mathematician. He published high-level academic articles and books, not only on mathematics, but also on physics, history, philosophy, and pedagogy. Levi was a member of the Bologna Aca ...
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
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order ''n'';
* ''C''
12;
* ''C''
2n Ă— ''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 more ...
.
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 rename ...
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 fo ...
s. In the early 1970s, the work of GĂ©rard Ligozat,
Daniel Kubert,
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 John Tate may refer to:
* John Tate (mathematician) (1925–2019), American mathematician
* John Torrence Tate Sr. (1889–1950), American physicist
* John Tate (Australian politician) (1895–1977)
* John Tate (actor) (1915–1979), Australian act ...
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 an ...
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
*
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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