Faltings's theorem

Faltings's theorem

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

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

The conjecture was later generalized by replacing ${\displaystyle \mathbb {Q} }$ by any

number field

.

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
  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

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

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

, ${\displaystyle C}$ by an arbitrary subvariety of ${\displaystyle A}$, and ${\displaystyle \Gamma }$ by an arbitrary finite-rank subgroup of ${\displaystyle A}$ leads to the .

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

Notes

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

• MR 1093712
  .
• MR 1096426
  .
• Cornell, Gary;
  MR 0861969. → Contains an English translation of Faltings (1983)
• MR 0718935
  .
• Faltings, Gerd (1984). "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".
  MR 0732554
  .
• Faltings, Gerd (1991). "Diophantine approximation on abelian varieties".
  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.
  MR 1307396
  .
• MR 0222087
  .
• Hindry, Marc; Silverman, Joseph H. (2000). Diophantine geometry.
  MR 1745599
  . → Gives Vojta's proof of Faltings's Theorem.
• ISBN 3-540-61223-8
  .
• Lawrence, Brian; Venkatesh, Akshay (2020). "Diophantine problems and p-adic period mappings". Invent. Math. 221 (3): 893–999.
  doi:10.1007/s00222-020-00966-7
  .
• 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.
• 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.
• doi:10.1070/IM1968v002n05ABEH000723
  .
• Shafarevich, I. R. (1963). "Algebraic number fields". Proceedings of the International Congress of Mathematicians: 163–176.
• MR 1109352
  .