I am looking for algorithms on how to find rational points on an elliptic curve $$y^2 = x^3 + a x + b$$ where $a$ and $b$ are integers. Any sort of ideas on how to proceed are welcome. For example, how to find solutions in which the numerators and denominators are bounded, or how to find solutions with a randomized algorithm. Anything better than brute force is interesting.

Background: a student worked on the Mordell-Weil theorem and illustrated it on some simple examples of elliptic curves. She looked for rational points by brute force (I really mean *brute*, by enumerating all possibilities and trying them). As a continuation of the project she is now interested in smarter algorithms for finding rational points. A cursory search on Math Reviews did not find much.

guaranteedto find the rational points on an elliptic curve. Section 4.2 of Poonen's survey www-math.mit.edu/~poonen/papers/millennial.pdf briefly discusses what is known and has references. – Qiaochu Yuan Oct 13 '10 at 14:22