Faltings' theorem
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In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. It was eventually proved by Gerd Faltings in 1983, about six decades after the conjecture was made; it is now known as Faltings' theorem.
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[edit] Background
Suppose we are given an algebraic curve C defined over the rational numbers (that is, C is defined by polynomials with rational coefficients), and suppose further that C is non-singular (in this case that condition isn't a real restriction). How many rational points (points with rational coefficients) are on C?
The answer depends upon the genus g of the curve. As is common in number theory, there are three cases: g = 0, g = 1, and g greater than 1. The g = 0 case has been understood for a long time; Mordell solved the g = 1 case, and conjectured the result for the g greater than 1 case.
[edit] Statement of results
The complete result is this:
Let C be a non-singular algebraic curve over the rationals of genus g. Then the number of rational points on C may be determined as follows:
- Case g = 0: no points or infinitely many; C is handled as a conic section.
- Case g = 1: no points, or C is an elliptic curve with a finite number of rational points forming an abelian group of quite restricted structure, or an infinite number of points forming a finitely generated abelian group (Mordell's Theorem, the initial result of the Mordell-Weil theorem).
- Case g > 1: according to Mordell's conjecture, now Faltings' Theorem, only a finite number of points.
[edit] Proofs
Faltings' 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. Different proofs have been found by Vojta and Bombieri, applying rather different methods.
[edit] 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 Shafarevich conjecture that there are only finitely many isomorphism classes of curves of genus greater than zero over a fixed number field with good reduction outside a given finite set of places;
- The Isogeny theorem that abelian varieties with isomorphic Tate modules are isogenous.
The reduction of the Mordell conjecture to the Shafarevich conjecture was due to A. N. Parshin in 1970.[1]
[edit] Generalizations
Because of the Mordell-Weil theorem, Faltings' 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 has been proved.
Another higher-dimensional generalization of Faltings' theorem is the Bombieri-Lang conjecture that if X is a pseudo-canonical variety (i.e., variety of general type) over a number field k, then X(k) is not Zariski dense in X. Even more general conjectures have been put forth by Paul Vojta.
[edit] References
- Cornell, Gary; Silverman, Joseph H. (1986). Arithmetic geometry. New York: Springer. ISBN 0387963111. → Contains an English translation of Faltings (1983)
- Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae 73 (3): 349–366. doi:.
- Hindry, Marc; Silverman, Joseph H. (2000). Diophantine geometry. Graduate Texts in Mathematics. 201. Springer-Verlag. ISBN 0-387-98981-1. → Gives Vojta's proof of Falting's Theorem.
- S. Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 101–122. ISBN 3-540-61223-8.
- Bombieri, Enrico (1990). "The Mordell conjecture revisited". Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (4): 615–640.
