Faltings's theorem

Faltings's 1983 paper had as consequences a number of statements which had previously been conjectured:


A sample application of Faltings's theorem is to a weak form of Fermat's Last Theorem: for any fixed ${\displaystyle n\geq 4}$ there are at most finitely many primitive integer solutions (pairwise coprime solutions) to ${\displaystyle a^{n}+b^{n}=c^{n}}$, since for such ${\displaystyle n}$ the Fermat curve ${\displaystyle x^{n}+y^{n}=1}$ has genus greater than 1.

Generalizations[edit]

Because of the Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve ${\displaystyle C}$ with a finitely generated subgroup ${\displaystyle \Gamma }$ of an abelian variety ${\displaystyle A}$. Generalizing by replacing ${\displaystyle A}$ by a semiabelian variety, ${\displaystyle C}$ by an arbitrary subvariety of ${\displaystyle A}$, and ${\displaystyle \Gamma }$ by an arbitrary finite-rank subgroup of ${\displaystyle A}$ leads to the Mordell–Lang conjecture, which was proved in 1995 by McQuillan^[9] following work of Laurent, Raynaud, Hindry, Vojta, and Faltings.


Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if ${\displaystyle X}$ is a pseudo-canonical variety (i.e., a variety of general type) over a number field ${\displaystyle k}$, then ${\displaystyle X(k)}$ is not Zariski dense in ${\displaystyle X}$. Even more general conjectures have been put forth by Paul Vojta.


The Mordell conjecture for function fields was proved by Yuri Ivanovich Manin^[10] and by Hans Grauert.^[11] In 1990, Robert F. Coleman found and fixed a gap in Manin's proof.^[12]