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In number theory, 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 conjectured 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.
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