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Mordellic variety
In mathematics, a Mordellic variety is an algebraic variety which has only finitely many points in any finitely generated field. The terminology was introduced by Serge Lang 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: 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 of E, then X is Mordellic.
Lang's conjectures
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 groups 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.
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 maps 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
Lang, Serge (1986). "Hyperbolic and Diophantine analysis" (PDF). Bulletin of the American Mathematical Society 14 (2): 159–205. doi:10.1090/s0273-0979-1986-15426-1. Zbl 0602.14019.
Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8.
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