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In number theory, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. The conjecture was later generalized by replacing Q by any number field. It was proved by , and is now known as Faltings's theorem. ==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's Theorem, ''C'' has only a finite number of rational points. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Faltings's theorem」の詳細全文を読む スポンサード リンク
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