Background
Let C be a non-singular algebraic curve of genus g over Q. Then the set of rational points on C may be determined as follows:
- Case g = 0: no points or infinitely many; C is handled as a conic section.
- Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group (Mordell's Theorem, later generalized to the Mordell–Weil theorem). Moreover Mazur's torsion theorem restricts the structure of the torsion subgroup.
- Case g > 1: according to the Mordell conjecture, now Faltings' Theorem, C has only a finite number of rational points.
Read more about this topic: Faltings' Theorem
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