Consider an elliptic curve E defined over Q. Assume that the rank of E(Q) is >=2. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank = algebraic rank.) How do you construct a point of infinite order on E(Q)?
(If the rank were 1, then the Gross-Zagier construction would do the job. If the rank were 0, then, of course, there would be no such point.)
Implicit in a paper of Mazur and Swinnerton-Dyer ("Arithmetic of Weil curves", Invent. Math., 25, 1-61 (1974); see especially section 2.4) there is a construction that seems to work a positive proportion of the time, though not always. Here is what the construction would be according to my understanding: take a modular parametrisation phi:X_0(N)->E(C), consider its points of ramification on the imaginary axis (there is at least one), take the image phi(z) of one such point z; due to standard magic, X_0(N) has an algebraic model that makes phi into an algebraic map; the trace of phi(z) is a point of E(Q) that might be non-torsion, and sometimes is).
Has any further work been done on this? (In particular, has it been proven that this works infinitely often?) Are there any other constructions for which similar statements have been conjectured or proven?