Editing Faltings's theorem (section)

==Generalizations==
Because of the [[Mordell–Weil theorem]], Faltings's theorem can be reformulated as a statement about the intersection of a curve <math>C</math> with a finitely generated subgroup <math>\Gamma</math> of an abelian variety <math>A</math>. Generalizing by replacing <math>A</math> by a [[semiabelian variety]], <math>C</math> by an arbitrary subvariety of <math>A</math>, and <math>\Gamma</math> by an arbitrary finite-rank subgroup of <math>A</math> leads to the [[Mordell–Lang conjecture]], which was proved in 1995 by [[Michael McQuillan (mathematician)|McQuillan]]{{sfn|McQuillan|1995}} following work of Laurent, [[Michel Raynaud|Raynaud]], Hindry, [[Paul Vojta|Vojta]], and [[Gerd Faltings|Faltings]].

Another higher-dimensional generalization of Faltings's theorem is the [[Bombieri–Lang conjecture]] that if <math>X</math> is a [[pseudo-canonical variety]] (i.e., a variety of general type) over a number field <math>k</math>, then <math>X(k)</math> is not [[Zariski topology|Zariski]] [[dense set|dense]] in <math>X</math>. Even more general conjectures have been put forth by [[Paul Vojta]].

The Mordell conjecture for function fields was proved by [[Yuri Ivanovich Manin]]{{sfn|Manin|1963}} and by [[Hans Grauert]].{{sfn|Grauert|1965}} In 1990, [[Robert F. Coleman]] found and fixed a gap in Manin's proof.{{sfn|Coleman|1990}}