In mathematics, a Mordellic variety is an
algebraic variety
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 ...
which has only finitely many points in any finitely generated field. The terminology was introduced by
Serge Lang
Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...
to enunciate a
range of conjectures linking the geometry of varieties to their Diophantine properties.
Formal definition
Formally, let ''X'' be a variety defined over an
algebraically closed field of
characteristic zero
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
: hence ''X'' is defined over a finitely generated field ''E''. If the set of points ''X''(''F'') is finite for any
finitely generated field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''F'' of ''E'', then ''X'' is Mordellic.
Lang's conjectures
The ''special set'' for a
projective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...
''V'' is the
Zariski closure
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
of the union of the images of all non-trivial maps from
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
...
s into ''V''. Lang conjectured that the complement of the special set is Mordellic.
A variety is ''algebraically hyperbolic'' if the special set is empty. Lang conjectured that a variety ''X'' is Mordellic if and only if ''X'' is algebraically hyperbolic and that this is in turn equivalent to ''X'' being
pseudo-canonical.
For a complex algebraic variety ''X'' we similarly define the ''analytic special'' or ''exceptional set'' as the Zariski closure of the union of images of non-trivial
holomorphic map
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivat ...
s from C to ''X''. Brody's definition of a hyperbolic variety is that there are no such maps. Again, Lang conjectured that a hyperbolic variety is Mordellic and more generally that the complement of the analytic special set is Mordellic.
References
*
* {{cite book , first=Serge , last=Lang , authorlink=Serge Lang , title=Survey of Diophantine Geometry , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, year=1997 , isbn=3-540-61223-8
Diophantine geometry
Algebraic varieties