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Rational points on elliptic curves John Tate, Joseph H. Silverman
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K

Moreover, it is a unirational variety: it admits a dominant rational map from a projective space. In other words, it is a two-sheeted cover of {\mathbb{P}^1} , and the sheets come together at {2g + 2} points. Mordell-Weil group and the central values of L-Series arsing from counting rational points over finite fields. The key to a conceptual proof of Lemma 1 is This point serves as the identity for a group law defined on any elliptic curve, which comes abstractly from an identification of an elliptic curve with its Jacobian variety. Rational Points on Modular Elliptic Curves book download Download Rational Points on Modular Elliptic Curves Request a Print Examination Copy. Similarly, if P is constrained to lie on one of the sides of the square, it becomes equivalent to showing that there are no non-trivial rational points on the elliptic curve y^2 = x^3 - 7x - 6 . Elliptic Curves, Modular Forms,. Rational Points on Elliptic Curves (Undergraduate Texts in Mathematics) By Joseph H. Is precisely the group of biholomorphic automorphisms of the Riemann sphere, which follows from the fact that the only meromorphic functions on the Riemann sphere are the rational functions. For elliptic curves, one has the Birch and Swinner-Dyer(BSD) conjecture which related the. Since it is a degree two cover, it is necessarily Galois, and {C} has a hyperelliptic involution {\iota: C \rightarrow C} over {\mathbb{ P}^1} with those is an elliptic curve (once one chooses an origin on {C} ), and the hyperelliptic .