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

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. The Zariski topology on Additional topics. The problem is therefore reduced to proving some curve has no rational points. In other words, it is a two-sheeted cover of {\mathbb{P}^1} , and the sheets come together at {2g + 2} points. Moreover, it is a unirational variety: it admits a dominant rational map from a projective space. This number depends only on the Kodaira symbol of the Jacobian and on an auxiliary rational point. Affine space and the Zariski topology; Regular functions; Regular maps. Possibilities include the 27 lines on a cubic surface, or an introduction to elliptic curves. If time permits, additional topics may be covered. 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 . Count the number of minimisations of a genus one curve defined over a Henselian discrete valuation field. We perform explicit computations on the special fibers of minimal proper regular models of elliptic curves. Rational functions and rational maps; Quasiprojective varieties. Whose rational points are precisely isomorphism classes of elliptic curves over {{\mathbb Q}} together with a rational point of order 13. Rational curves; Relation with field theory; Rational maps; Singular and nonsingular points; Projective spaces. The secant procedure allows one to define a group structure on the set of rational points on a elliptic curves (that is, points whose coordinates are rational numbers). The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study.