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

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 . If two points P, Q on an elliptic curve have rational coordinates then so does P*Q. 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. In other words, it is a two-sheeted cover of {\mathbb{P}^1} , and the sheets come together at {2g + 2} points. In particular, you can take Q=P, so that the line PQ is the tangent at P. We perform explicit computations on the special fibers of minimal proper regular models of elliptic curves. Then there is a constant B(d) depending only on d such that, if E/K is an elliptic curve with a K -rational torsion point of order N , then N < B(d) . Theorem (Uniform Boundedness Theorem).Let K be a number field of degree d . This number depends only on the Kodaira symbol of the Jacobian and on an auxiliary rational point. Moreover, it is a unirational variety: it admits a dominant rational map from a projective space. Count the number of minimisations of a genus one curve defined over a Henselian discrete valuation field. 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 . Rational Points on Elliptic Curves John Tate (Auteur), J.H.