Rational Points on Elliptic CurvesThe theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. "Rational Points on Elliptic Curves" stresses this interplay as it develops the basic theory, thereby providing an opportunity for advance undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make "Rational Points on Elliptic Curves" an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry. |
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abelian group algebraic algorithm auxiliary polynomial ax² b₁ Bezout's theorem C₁ C1 and C2 C₂ complex numbers complex points compute conic cubic curve cubic equation curve y² cyclic group defined denominator Diophantine elliptic curve example Fermat's finite number finite order Galois extension Galois group geometry gives group law group of order group of rational height Hence homogeneous homomorphism inequality integer coefficients integer coordinates integer points integer solutions isomorphism Lemma linear look matrix mod Q*2 modulo Mordell's theorem Nagell-Lutz theorem number theory one-to-one order dividing order three P₁ point at infinity points of finite points of order polynomial f(x projective curve projective plane prove quadratic rational numbers rational points real numbers roots satisfy Section singular cubic subgroup Suppose tangent line Weierstrass equation