Let be a number field and let be a non-singular curve defined over and genus . When the genus is , the curve is isomorphic to (over an algebraic closure ) and therefore is either empty or equal to (in particular is infinite). If the genus of is and contains at least one point over then is an elliptic curve and the Mordell-Weil theorem shows that is a finitely generated abelian group (in particular, may be finite or infinite). However, if , Mordell conjectured in that cannot be infinite. This was first proven by Faltings in .
Theorem (Faltings’ Theorem (Mordell’s conjecture)).
Let be a number field and let be a non-singular curve defined over of genus . Then is finite.
The reader may also be interested in Siegel’s theorem.
|Date of creation||2013-03-22 15:57:21|
|Last modified on||2013-03-22 15:57:21|
|Last modified by||alozano (2414)|