| Mordell Conjecture |
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Information AboutMordell Conjecture |
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BACKGROUND Suppose we are given an Algebraic Curve ''C'' defined over the Rational Number s (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 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:
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 Model s. A number of subsequent proofs have since been found, applying rather different methods. |
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