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Rational points on elliptic curves by John Tate, Joseph H. Silverman

Rational points on elliptic curves John Tate, Joseph H. Silverman ebook
ISBN: 3540978259, 9783540978251
Page: 296
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Format: djvu

In other words, it is a two-sheeted cover of {\mathbb{P}^1} , and the sheets come together at {2g + 2} points. Count the number of minimisations of a genus one curve defined over a Henselian discrete valuation field. Moreover, it is a unirational variety: it admits a dominant rational map from a projective space. Ratpoints (C library): Michael Stoll's highly optimized C program for searching for certain rational points on hyperelliptic curves (i.e. It also has It has no dependencies (instead of PARI), because Mark didn't want to have to license sympow under the GPL. The concrete example he described, which had been the original question of Masser, was the following: consider the Legendre family of elliptic curves. 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 . We perform explicit computations on the special fibers of minimal proper regular models of elliptic curves. Mordell-Weil group and the central values of L-Series arsing from counting rational points over finite fields. This number depends only on the Kodaira symbol of the Jacobian and on an auxiliary rational point. This library is very, very good and fast for doing computations of many functions relevant to number theory, of "class groups of number fields", and for certain computations with elliptic curves. Smyth, Minimal polynomials of algebraic numbers with rational parameters. For elliptic curves, one has the Birch and Swinner-Dyer(BSD) conjecture which related the. Is a smooth projective curve of genus 1 (i.e., topologically a torus) defined over {K} with a {K} -rational point {0} . Silverman, Lehmer's Conjecture and points on elliptic curves that are congruent to torsion points. P_t=(2,p_t),\quad Q_t=(3,q_t These techniques are quite novel in this area, and rely ultimately (and quite strikingly) on the circle of ideas that started with the 1989 work of Bombieri and Pila on the number of rational (or integral) points on transcendental curves (in the plane, say). We give geometric criteria which relate these models to the minimal proper regular models of the Jacobian elliptic curves of the genus one curves above.