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Faltings' TheoremIn 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 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 an non-singular algebraic curve over the rationals of genus g. Then the number of rational points on C may be determined as follows: 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 Nron models. A number of subsequent proofs have since been found, applying rather different methods.
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