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

In 1922 Louis Mordell proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis. This number depends only on the Kodaira symbol of the Jacobian and on an auxiliary rational point. An upper bound is established for certain exponential sums on the rational points of an elliptic curve over a residue class ring ZN , N=pq for two distinct odd primes p and q. The problem is therefore reduced to proving some curve has no rational points. Count the number of minimisations of a genus one curve defined over a Henselian discrete valuation field. Henri Poincaré studied them in the early years of the 20th century. We perform explicit computations on the special fibers of minimal proper regular models of elliptic curves. Sub Child Category 1; Sub Child Category 2; Sub Child Category 3. Whose rational points are precisely isomorphism classes of elliptic curves over {{\mathbb Q}} together with a rational point of order 13. Elliptic curves have been a focus of intense scrutiny for decades. K3 surfaces, level structure, and rational points. 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. Is a smooth projective curve of genus 1 (i.e., topologically a torus) defined over {K} with a {K} -rational point {0} . 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. It had long been known that the rational points on an elliptic curve, defined over the rationals, form a group Γ under a chord and tangent construction; Mordell proved that Γ has a finite basis. Moduli spaces of elliptic curves with level structure are fundamental for arithmetic and Diophantine problems over number fields in particular.