Faltings' theorem

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.

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 > 1. The g = 0 case has been understood for a long time; Mordell solved the g = 1 case, and conjectured the result when g > 1.

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.

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.

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]. A sample application of Falting's theorem is to Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive solutions to aⁿ + bⁿ = cⁿ (since the curve xⁿ + yⁿ = 1 has genus greater than 1).

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.

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:10.1007/BF01388432. 
• 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. 

Notes