Faltings's theorem

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. In 1983 it was proved by Gerd Faltings (1983, 1984), and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field.

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.

Proofs

Faltings's 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 Vojta (1991). A more elementary variant of Vojta's proof was given by Bombieri (1990).

Consequences

Faltings's 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 Q_l-modules with Galois action) are isogenous.

The reduction of the Mordell conjecture to the Shafarevich conjecture was due to A. N. Paršin (1971). A sample application of Faltings's 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.

Generalizations

Because of the Mordell–Weil theorem, Faltings's 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 was proved by Faltings (1991, 1994).

Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if X is a pseudo-canonical variety (i.e., a 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.

The Mordell conjecture for function fields was proved by Manin (1963) and by Grauert (1965). In 1990, Coleman (1990) found and fixed a gap in Manin's proof.

References

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• Coleman, Robert F. (1990). "Manin's proof of the Mordell conjecture over function fields". L'Enseignement Mathématique. Revue Internationale. IIe Série. 36 (3): 393–427. ISSN 0013-8584. MR 1096426. Archived from the original on 2011-10-02.
• Cornell, Gary; Silverman, Joseph H., eds. (1986). Arithmetic geometry. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30 – August 10, 1984. New York: Springer-Verlag. doi:10.1007/978-1-4613-8655-1. ISBN 0-387-96311-1. MR 0861969. → Contains an English translation of Faltings (1983)
• Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields]. Inventiones Mathematicae (in German). 73 (3): 349–366. doi:10.1007/BF01388432. MR 0718935.
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• Faltings, Gerd (1991). "Diophantine approximation on abelian varieties". Ann. of Math. 133 (3): 549–576. doi:10.2307/2944319. MR 1109353.
• Faltings, Gerd (1994). "The general case of S. Lang's conjecture". In Cristante, Valentino; Messing, William. Barsotti Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991. Perspectives in Mathematics. San Diego, CA: Academic Press, Inc. ISBN 0-12-197270-4. MR 1307396.
• Grauert, Hans (1965). "Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper". Publications Mathématiques de l'IHÉS (25): 131–149. ISSN 1618-1913. MR 0222087.
• Hindry, Marc; Silverman, Joseph H. (2000). Diophantine geometry. Graduate Texts in Mathematics. 201. New York: Springer-Verlag. doi:10.1007/978-1-4612-1210-2. ISBN 0-387-98981-1. MR 1745599. → Gives Vojta's proof of Faltings's Theorem.
• Lang, Serge (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 101–122. ISBN 3-540-61223-8.
• Manin, Ju. I. (1963). "Rational points on algebraic curves over function fields". Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya (in Russian). 27: 1395–1440. ISSN 0373-2436. MR 0157971. (Translation: Manin, Yu. (1966). "Rational points on algebraic curves over function fields". American Mathematical Society Translations: Series 2. 59: 189–234. ISSN 0065-9290. )
• Mordell, Louis J. (1922). "On the rational solutions of the indeterminate equation of the third and fourth degrees". Proc. Cambridge Philos. Soc. 21: 179–192.
• Paršin, A. N. (1970). "Quelques conjectures de finitude en géométrie diophantienne" (PDF). Actes du Congrès International des Mathématiciens. Tome 1. Nice: Gauthier-Villars (published 1971). pp. 467–471. MR 0427323. Archived from the original (PDF) on 2016-09-24. Retrieved 2016-06-11.
• Parshin, A. N. (2001) [1994], "Mordell conjecture", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Vojta, Paul (1991). "Siegel's theorem in the compact case". Ann. of Math. 133 (3): 509–548. doi:10.2307/2944318. MR 1109352.