Editing Faltings's theorem (section)

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

* The ''Mordell conjecture'' that a curve of genus greater than 1 over a number field has only finitely many rational points;
* The ''Isogeny theorem'' that abelian varieties with isomorphic [[Tate module]]s (as <math>\mathbb{Q}_{\ell}</math>-modules with Galois action) are [[Isogeny|isogenous]].

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