Faltings's theorem

Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field ${\displaystyle \mathbb {Q} }$ of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell,^[1] and known as the Mordell conjecture until its 1983 proof by Gerd Faltings.^[2] The conjecture was later generalized by replacing ${\displaystyle \mathbb {Q} }$ by any number field.

Faltings's theorem

Gerd Faltings
Field	Arithmetic geometry
Conjectured by	Louis Mordell
Conjectured in	1922
First proof by	Gerd Faltings
First proof in	1983
Generalizations	Bombieri–Lang conjecture Mordell–Lang conjecture
Consequences	Siegel's theorem on integral points

Background

Let ${\displaystyle C}$  be a non-singular algebraic curve of genus ${\displaystyle g}$  over ${\displaystyle \mathbb {Q} }$ . Then the set of rational points on ${\displaystyle C}$  may be determined as follows:

• When ${\displaystyle g=0}$ , there are either no points or infinitely many. In such cases, ${\displaystyle C}$  may be handled as a conic section.
• When ${\displaystyle g=1}$ , if there are any points, then ${\displaystyle C}$  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 ${\displaystyle g>1}$ , according to Faltings's theorem, ${\displaystyle C}$  has only a finite number of rational points.

Proofs

Igor Shafarevich conjectured 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 fixed finite set of places.^[3] Aleksei Parshin showed that Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick.^[4]

Gerd Faltings proved Shafarevich's finiteness conjecture using a known reduction to a case of the Tate conjecture, together with tools from algebraic geometry, including the theory of Néron models.^[5] The main idea of Faltings's proof is the comparison of Faltings heights and naive heights via Siegel modular varieties.^[a]

Later proofs

• Paul Vojta gave a proof based on diophantine approximation.^[6] Enrico Bombieri found a more elementary variant of Vojta's proof.^[7]
• Brian Lawrence and Akshay Venkatesh gave a proof based on p-adic Hodge theory, borrowing also some of the easier ingredients of Faltings's original proof.^[8]

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 Isogeny theorem that abelian varieties with isomorphic Tate modules (as ${\displaystyle \mathbb {Q} _{\ell }}$ -modules with Galois action) are isogenous.

A sample application of Faltings's theorem is to a weak form of Fermat's Last Theorem: for any fixed ${\displaystyle n\geq 4}$  there are at most finitely many primitive integer solutions (pairwise coprime solutions) to ${\displaystyle a^{n}+b^{n}=c^{n}}$ , since for such ${\displaystyle n}$  the Fermat curve ${\displaystyle x^{n}+y^{n}=1}$  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 ${\displaystyle C}$  with a finitely generated subgroup ${\displaystyle \Gamma }$  of an abelian variety ${\displaystyle A}$ . Generalizing by replacing ${\displaystyle A}$  by a semiabelian variety, ${\displaystyle C}$  by an arbitrary subvariety of ${\displaystyle A}$ , and ${\displaystyle \Gamma }$  by an arbitrary finite-rank subgroup of ${\displaystyle A}$  leads to the Mordell–Lang conjecture, which was proved in 1995 by McQuillan^[9] following work of Laurent, Raynaud, Hindry, Vojta, and Faltings.

Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if ${\displaystyle X}$  is a pseudo-canonical variety (i.e., a variety of general type) over a number field ${\displaystyle k}$ , then ${\displaystyle X(k)}$  is not Zariski dense in ${\displaystyle X}$ . Even more general conjectures have been put forth by Paul Vojta.

The Mordell conjecture for function fields was proved by Yuri Ivanovich Manin^[10] and by Hans Grauert.^[11] In 1990, Robert F. Coleman found and fixed a gap in Manin's proof.^[12]

Notes

1. "Faltings relates the two notions of height by means of the Siegel moduli space.... It is the main idea of the proof." Bloch, Spencer (1984). "The Proof of the Mordell Conjecture". The Mathematical Intelligencer. 6 (2): 44. doi:10.1007/BF03024155. S2CID 306251.

Citations

1. Mordell 1922.
2. Faltings 1983; Faltings 1984.
3. Shafarevich 1963.
4. Parshin 1968.
5. Faltings 1983.
6. Vojta 1991.
7. Bombieri 1990.
8. Lawrence & Venkatesh 2020.
9. McQuillan 1995.
10. Manin 1963.
11. Grauert 1965.
12. Coleman 1990.

References

• Bombieri, Enrico (1990). "The Mordell conjecture revisited". Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (4): 615–640. MR 1093712.
• Coleman, Robert F. (1990). "Manin's proof of the Mordell conjecture over function fields". L'Enseignement Mathématique. 2e Série. 36 (3): 393–427. ISSN 0013-8584. MR 1096426.
• 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. Bibcode:1983InMat..73..349F. doi:10.1007/BF01388432. MR 0718935.
• Faltings, Gerd (1984). "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae (in German). 75 (2): 381. doi:10.1007/BF01388572. MR 0732554.
• Faltings, Gerd (1991). "Diophantine approximation on abelian varieties". Ann. of Math. 133 (3): 549–576. doi:10.2307/2944319. JSTOR 2944319. MR 1109353.
• Faltings, Gerd (1994). "The general case of S. Lang's conjecture". In Cristante, Valentino; Messing, William (eds.). 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 (25): 131–149. doi:10.1007/BF02684399. ISSN 1618-1913. MR 0222087.
• Hindry, Marc; Silverman, Joseph H. (2000). Diophantine geometry. Graduate Texts in Mathematics. Vol. 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.
• Lawrence, Brian; Venkatesh, Akshay (2020). "Diophantine problems and p-adic period mappings". Invent. Math. 221 (3): 893–999. arXiv:1807.02721. doi:10.1007/s00222-020-00966-7.
• 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. doi:10.1090/trans2/050/11. ISBN 9780821817506. ISSN 0065-9290. )
• McQuillan, Michael (1995). "Division points on semi-abelian varieties". Invent. Math. 120 (1): 143–159. doi:10.1007/BF01241125.
• 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). (PDF). Actes du Congrès International des Mathématiciens. Vol. 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". Encyclopedia of Mathematics. EMS Press.
• Parshin, A. N. (1968). "Algebraic curves over function fields I". Izv. Akad. Nauk. SSSR Ser. Math. 32 (5): 1191–1219. Bibcode:1968IzMat...2.1145P. doi:10.1070/IM1968v002n05ABEH000723.
• Shafarevich, I. R. (1963). "Algebraic number fields". Proceedings of the International Congress of Mathematicians: 163–176.
• Vojta, Paul (1991). "Siegel's theorem in the compact case". Ann. of Math. 133 (3): 509–548. doi:10.2307/2944318. JSTOR 2944318. MR 1109352.