In number theory, the Mordell conjecture is the conjecture made by Mordell (1922) 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 a finite extension. It was proved by Gerd Faltings (1983), and is now known as Faltings' theorem.
- 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.
Faltings' original proof used the known reduction to a case of the Tate conjecture, and a number of tools from algebraic geometry, including the theory of Néron models. A very different proof, based on diophantine approximation, was found by Paul Vojta. A more elementary variant of Vojta's proof was given by Enrico Bombieri.
Faltings' 1983 paper had as consequences a number of statements which had previously been conjectured:
- The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points;
- The Shafarevich conjecture that there are only finitely many isomorphism classes of abelian varieties of fixed dimension and fixed polarization degree over a fixed number field with good reduction outside a given finite set of places; and
- The Isogeny theorem that abelian varieties with isomorphic Tate modules (as Ql-modules with Galois action) are isogenous.
The reduction of the Mordell conjecture to the Shafarevich conjecture was due to Parshin (1971). A sample application of Faltings' theorem is to a weak form of Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive integer solutions to an + bn = cn, since for such n the curve xn + yn = 1 has genus greater than 1.
Because of the Mordell–Weil theorem, Faltings' theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell–Lang conjecture, which has been proved.
Another higher-dimensional generalization of Faltings' theorem is the Bombieri–Lang conjecture that if X is a pseudo-canonical variety (i.e., variety of general type) over a number field k, then X(k) is not Zariski dense in X. Even more general conjectures have been put forth by Paul Vojta.
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