Faltings's theorem


In arithmetic geometry, the Mordell conjecture is the conjecture made by 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, 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:
posed a finiteness conjecture that asserted 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. showed that the Mordell conjecture would hold if the Shafarevich finiteness conjecture was true using Parshin's trick.
proved the Shafarevich finiteness conjecture using a known reduction to a case of the Tate conjecture, and a number of tools from algebraic geometry, including the theory of Néron models. The main idea of Faltings' proof is the comparison of Faltings heights and naive heights via Siegel modular varieties.

Later proofs

A proof based on diophantine approximation was given by. A more elementary variant of Vojta's proof was given by.

Consequences

Faltings's 1983 paper had as consequences a number of statements which had previously been conjectured:
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.
Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if X is a pseudo-canonical variety over a number field k, then X 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 and by. In 1990, found and fixed a gap in Manin's proof.

Footnotes