Mordellic variety

2020 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