# Mordellic variety

2010 Mathematics Subject Classification: Primary: 11G [MSN][ZBL]

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.

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/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 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 \$\mathbb{C}\$ to \$X\$. Brody's definition of a hyperbolic variety is that there are no such maps: cf Kobayashi hyperbolicity. 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 Zbl 0602.14019.
• Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8
How to Cite This Entry:
Mordellic variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mordellic_variety&oldid=35769