In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Vojta's conjecture is a conjecture introduced by about heights of points on
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
over
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 ...
s. The conjecture was motivated by an analogy between
diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by r ...
and
Nevanlinna theory In the mathematical field of complex analysis, Nevanlinna theory is part of the
theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl called it "one of the few great mathematical events of (the twentieth) century ...
(value distribution theory) in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
. It implies many other conjectures in
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by r ...
,
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
s,
arithmetic geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic variety, alg ...
, and
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
.
Statement of the conjecture
Let
be a number field, let
be a non-singular algebraic variety, let
be an effective
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
on
with at worst normal crossings, let
be an ample divisor on
, and let
be a canonical divisor on
. Choose Weil
height function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic ...
s
and
and, for each
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
on
, a local height function
. Fix a finite set of absolute values
of
, and let
. Then there is a constant
and a non-empty Zariski open set
, depending on all of the above choices, such that
::
Examples:
# Let
. Then
, so Vojta's conjecture reads
for all
.
# Let
be a variety with trivial canonical bundle, for example, an
abelian variety
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 func ...
, a
K3 surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected alg ...
or a
Calabi-Yau variety
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring ...
. Vojta's conjecture predicts that if
is an effective ample normal crossings divisor, then the
-integral points on the affine variety
are not Zariski dense. For abelian varieties, this was conjectured by
Lang
Lang may refer to:
*Lang (surname), a surname of independent Germanic or Chinese origin
Places
* Lang Island (Antarctica), East Antarctica
* Lang Nunatak, Antarctica
* Lang Sound, Antarctica
* Lang Park, a stadium in Brisbane, Australia
* Lang, ...
and proven by .
# Let
be a variety of
general type In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''.
Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
, i.e.,
is ample on some non-empty Zariski open subset of
. Then taking
, Vojta's conjecture predicts that
is not Zariski dense in
. This last statement for varieties of general type is the
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 ...
.
Generalizations
There are generalizations in which
is allowed to vary over
, and there is an additional term in the upper bound that depends on the discriminant of the field extension
.
There are generalizations in which the non-archimedean local heights
are replaced by truncated local heights, which are local heights in which multiplicities are ignored. These versions of Vojta's conjecture provide natural higher-dimensional analogues of the
ABC conjecture
The ''abc'' conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b' ...
.
References
*
*{{cite journal , last1=Faltings , first1=Gerd , author1-link=Gerd Faltings , title=Diophantine approximation on abelian varieties , journal=Annals of Mathematics , mr=1109353, year=1991 , volume=123 , pages=549–576 , doi=10.2307/2944319 , issue=3
Conjectures
Unsolved problems in number theory