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Concept
Faltings's theorem
Summary
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field 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 by any number field.
Let be a non-singular algebraic curve of genus over . Then the set of rational points on may be determined as follows:
When , there are either no points or infinitely many. In such cases, may be handled as a conic section.
When , if there are any points, then 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 , according to Faltings's theorem, has only a finite number of rational points.
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.
Paul Vojta gave a proof based on diophantine approximation. Enrico Bombieri found a more elementary variant of Vojta's proof.
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.
Official source
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