Editing Faltings's theorem (section)

{{Short description|Curves of genus > 1 over the rationals have only finitely many rational points}}
{{Infobox mathematical statement
| name = Faltings's theorem
| image = Gerd Faltings MFO.jpg
| caption = Gerd Faltings
| field = [[Arithmetic geometry]]
| conjectured by = [[Louis Mordell]]
| conjecture date = 1922
| first proof by = [[Gerd Faltings]]
| first proof date = 1983
| open problem = 
| known cases =
| implied by =
| equivalent to = 
| generalizations = [[Bombieri–Lang conjecture]]<br/>[[Glossary of arithmetic and diophantine geometry#M|Mordell–Lang conjecture]]
| consequences = [[Siegel's theorem on integral points]]
}}

'''Faltings's theorem''' is a result in [[arithmetic geometry]], according to which a curve of [[Genus (mathematics)|genus]] greater than 1 over the field <math>\mathbb{Q}</math> of [[rational number]]s has only finitely many [[rational point]]s. This was conjectured in 1922 by [[Louis Mordell]],{{sfn|Mordell|1922}} and known as the '''Mordell conjecture''' until its 1983 proof by [[Gerd Faltings]].{{sfnm|1a1=Faltings|2a1=Faltings|1y=1983|2y=1984}} The conjecture was later generalized by replacing <math>\mathbb{Q}</math> by any [[number field]].