The conjecture, eventually proved by
Gerd Faltings[?] after about six decades, states a basic result on rational number solutions to
Diophantine equations[?].
From the point of view of number theory, the classification of algebraic curves[?] that matters is into three classes, according to their genus g. Suppose given such a curve C defined over the rational numbers (that is, by polynomials with rational coefficients) and non-singular[?] (in this case that condition isn't a real restriction). Then there are three cases, according to how many rational points (points with rational coefficients) are on C
- Case g = 0 : no points or infinitely many, C is handled as a conic section.
- Case g = 1: no points, or an elliptic curve with a finite number forming an abelian group of quite restricted structure, or an infinite number 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.
Therefore the conjecture took its natural place in the overall picture.
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