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:

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).

Faltings's 1983 paper had as consequences a number of statements which had previously been conjectured:

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 *a*^{n} + *b*^{n} = *c*^{n}, since for such *n* the curve *x*^{n} + *y*^{n} = *1* has genus greater than 1.

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).

This page was last edited on 21 December 2017, at 19:59.

Reference: https://en.wikipedia.org/wiki/Faltings%27_theorem under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Faltings%27_theorem under CC BY-SA license.

- Number Theory
- Mordell
- 1922
- Genus
- Rational Points
- Gerd Faltings
- 1983
- 1984
- Number Field
- Non-singular
- Genus
- Tate Conjecture
- Algebraic Geometry
- NÃ©ron Models
- Vojta
- 1991
- Bombieri
- 1990
- A. N. ParÅ¡in
- 1971
- Fermat's Last Theorem
- MordellâWeil Theorem
- MordellâLang Conjecture
- 1991
- 1994

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