Computability of rational points on elliptic curves (rank computability)

Determine whether there exists a finite-time algorithm (a Turing machine that terminates on all inputs) that, given an elliptic curve E/ℚ, outputs the set of rational points E(ℚ), equivalently whether the rank of an elliptic curve over ℚ is a computable quantity.

Background

The authors contrast their conditional algorithms for hyperbolic curves with the current state for elliptic curves: while procedures exist that would compute E(ℚ) assuming standard conjectures, it is unknown unconditionally whether a terminating algorithm exists for all inputs.

This question is intimately tied to the computability of the Mordell–Weil rank and finiteness of the Tate–Shafarevich group.