Curves of genus ; 1 over the rationals have only finitely many rational points
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field formula_1 of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. The conjecture was later generalized by replacing formula_1 by any number field.
Background.
Let formula_3 be a non-singular algebraic curve of genus formula_4 over formula_1. Then the set of rational points on formula_3 may be determined as follows:
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. Aleksei Parshin showed that Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick. 
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. The main idea of Faltings's proof is the comparison of Faltings heights and naive heights via Siegel modular varieties.
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 formula_14 there are at most finitely many primitive integer solutions (pairwise coprime solutions) to formula_15, since for such formula_16 the Fermat curve formula_17 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 formula_3 with a finitely generated subgroup formula_19 of an abelian variety formula_20. Generalizing by replacing formula_20 by a semiabelian variety, formula_3 by an arbitrary subvariety of formula_20, and formula_19 by an arbitrary finite-rank subgroup of formula_20 leads to the Mordell–Lang conjecture, which was proved in 1995 by McQuillan following work of Laurent, Raynaud, Hindry, Vojta, and Faltings.
Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if formula_26 is a pseudo-canonical variety (i.e., a variety of general type) over a number field formula_27, then formula_28 is not Zariski dense in formula_26. Even more general conjectures have been put forth by Paul Vojta.
The Mordell conjecture for function fields was proved by Yuri Ivanovich Manin and by Hans Grauert. In 1990, Robert F. Coleman found and fixed a gap in Manin's proof.
Citations.
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References.
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