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

An elliptic curve E defined over a finite field F is a plane non-singular cubic curve with at least a rational point [10]. I compare this book to Rational Points on Elliptic Curves (RP) by Tate and Silverman, and The Arithmetic of Ellipitic Curves (AEC) by Silverman. 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 . This process never repeats itself (and so infinitely many rational points may be generated in this way). Solid intermediate introduction to elliptic curves. After a nice work lunch with two of my soon-to-be collaborators, I attended Wei Ho's talk in the Current Events Bulletin on “How many rational points does a random curve have?”. 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. Rational Points on Elliptic Curves - Silverman, Tate.pdf. Wei Ho delivered a very Ho talked about how Bhargava and his school are approaching different conjectures on the ranks of elliptic curves. Hey, now we know that this is a question in arithmetic statistics! In the language of elliptic curves, given a rational point P we are considering the new rational point -2P . Elliptic Curve Cryptography and ECIES. Points on elliptic curves over Q which are not [0:1:0] have their last coordinate =1 but sometimes this is an int (not even an Integer) which breaks some code: sage: E=EllipticCurve('37a1') sage: [type(c) for c in E(0)] [