View source for Mordellic variety

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An [[algebraic variety]] which has only finitely many points in any finitely generated field.  The terminology was introduced by Serge Lang to enunciate a [[Lang conjecture on analytically hyperbolic varieties|range of conjectures]] linking the geometry of varieties to their Diophantine properties.

Formally, let $X$ be a variety defined over an [[algebraically closed field]] of [[Characteristic of a field|characteristic zero]]: 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]] $F/E$, then $X$ is Mordellic.

The ''special set'' for a [[projective variety]] $V$ is the [[Zariski closure]] of the union of the images of all non-trivial maps from [[algebraic group]]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 turn equivalent to $X$ being [[pseudo-canonical variety|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]]s from $\mathbb{C}$ 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====
* Lang, Serge (1986). "Hyperbolic and Diophantine analysis". Bulletin of the American Mathematical Society 14 (2): 159–205. {{DOI|10.1090/s0273-0979-1986-15426-1}} {{DOI|0602.14019}}.
* Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8