The Mordell conjecture states 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.
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 (though 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 conjected the result for the g greater than 1 case.
The complete result is this:
Let C be an 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 = 2: according to Mordell's conjecture, now Faltings' Theorem, only a finite number of points.
