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§6. Rational Points on Rational Curves. Faltings and the Mordell Conjecture  17

        §6. Rational Points on Rational Curves. Faltings and the Mordell
            Conjecture

        The cases of rational points on curves of degrees 1, 2, and 3 have been considered,
        and we were led naturally into the study of elliptic curves by our simple geomet-
        ric approach to these diophantine equations. Before going into elliptic curves, we
        mention some things about curves of degree strictly greater than 3.
        (6.1) Mordell Conjecture (For Plane Curves). Let C be a smooth rational plane
        curve of degree strictly greater than 3. Then the set C(Q) of rational points on C is
        finite.
           This conjecture was proved by Faltings in 1983 and is a major achievement in
        diophantine geometry to which many mathematicians have contributed. Some of the
        ideas in the proof simplify known results for elliptic curves and we will come back
        to the subject later.
           For curves other than lines, conics, and cubics, it is often necessary to consider
        models of the curve in higher dimensions and with more than one equation. This
        leads one directly into algebraic geometry and general notions of algebraic varieties.
        The topics in elliptic curves treated in detail in this book are exactly those which use
        only a minimum of algebraic geometry, namely the theory of plane curves given in
        2.
           From a descriptive point of view the complex points X(C) of an algebraic curve
        defined over the complex numbers C have a local structure since X(C) is homeomor-
        phic to an open disc in the complex plane with change of variable given by analytic
        functions. Topologically X(C) is a closed oriented surface with some number of g
        holes.
















        (6.2) Definition. The invariant g is called the genus of the curve.
           There are algebraic formulations of the notion of genus, and it is a well-defined
        quantity associated with any algebraic curve. Lines and conics have genus g = 0,
        singular cubics have genus g = 0, and nonsingular cubics have genus g = 1.
        (6.3) Assertion. A nonsingular plane curve of degree d has genus
                                     (d − 2)(d − 1)
                                 g =              .
                                           2
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