Editing Faltings's theorem (section)

==Background==
Let <math>C</math> be a [[Singular point of a curve|non-singular]] algebraic curve of [[genus (mathematics)|genus]] <math>g</math> over <math>\mathbb{Q}</math>. Then the set of rational points on <math>C</math> may be determined as follows:
* When <math>g=0</math>, there are either no points or infinitely many. In such cases, <math>C</math> may be handled as a [[conic section]].
* When <math>g=1</math>, if there are any points, then <math>C</math> is an [[elliptic curve]] and its rational points form a [[finitely generated abelian group]]. (This is ''Mordell's Theorem'', later generalized to the [[Mordell–Weil theorem]].) Moreover, [[Mazur's torsion theorem]] restricts the structure of the torsion subgroup.
* When <math>g>1</math>, according to Faltings's theorem, <math>C</math> has only a finite number of rational points.