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

One reason for interest in the BSD conjecture is that the Clay Mathematics Institute is of a rational parametrization which is introduced on page 10. Position: Location: Field of Science: Science - Math - Number Theory - Elliptic Curves and Modular Forms. Theorem 5 (on page vi) of Diem's thesis states that the discrete logarithm problem in the group of rational points of an elliptic curves E( F_{p^n} ) can be solved in an expected time of \tilde{O}( q^{2 – 2/n} ) bit operations. Website / Blog: www.math.rutgers.edu/~tunnell/math574.html. The book surveys some recent developments in the arithmetic of modular elliptic curves. A little more difficult, I really enjoyed Silverman+Tate's Rational Points on Elliptic Curves and Stewart+Tall's Algebraic Number Theory. Akhil Mathew - August 17, 2009. This is precisely to look for rational points on the modular surface S parametrizing pairs (E,E',C,C',φ), where E and E' are elliptic curves, C and C' are cyclic 13-subgroups, and φ is an isomorphism between C and C'. In 1922 Louis Mordell proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis. The subtitle is: Curves, Counting, and Number Theory and it is an introduction to the theory of Elliptic curves taking you from an introduction up to the statement of the Birch and Swinnerton-Dyer (BSD) Conjecture. Some sample rational points are shown in the following graph. Sub Child Category 1; Sub Child Category 2; Sub Child Category 3. In the elliptic curve E: y^2+y=x^3-x , the rational points form a group of rank 1 (i.e., an infinite cyclic group), and can be generated by P =(0,0) under the group law. Name: Institution: Rational Points on Elliptic Curves. Update: also, opinions on books on elliptic curves solicited, for the four or five of you who might have some! If you're interested in algebraic geometry from an elementary point of view, Tate and Silverman's Rational Points on Elliptic Curves is also worth checking out. That is, an equation for a curve that provides all of the rational points on that curve. 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. Are (usually) three distinct groups of prime order p . The two groups G_1 and G_2 correspond to subgroups of K -rational points E(K) of an elliptic curve E over a finite field K with characteristic q different from p .