Faltings' theorem

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Categories: Number theory | Algebraic curves | Diophantine geometry | Mathematical theorems

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.

1 Background
2 Statement of results
3 Proofs
4 References

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.

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. A number of subsequent proofs have since been found, applying rather different methods.