TITLE: Which curves have infinitely many rational points
QUESTION [14 upvotes]: Question: Assuming finiteness of the Tate-Shafarevich group, is there an algorithm to determine whether a curve $C$ defined over a number field $K$ has infinitely many $K$-rational points?

I believe that this is (a) true and (b) sufficiently important that it has been carefully explained somewhere, but I don't know a reference. Any help from the MO community would be very much appreciated! 
P.S. To make the question precise, $C$ is specified, say, by its function field (a simple extension of $K(x)$), and all abelian varieties over all number fields have finite Sha. (And if the algorithm takes $10^{10}$ years for C: y=3x+1 over the rationals, I don't care.)

REPLY [7 votes]: Dear Tim: 
How  right you are to worry about computing the Jacobian of a genus one curve! Years ago, in the midst of many discussions with Bill McCallum, Alex Perlis, Nick Shepherd-Barron, and John Tate, I convinced myself this could be done, and it was eventually written up in Alex's Arizona thesis:
http://math.arizona.edu/~aprl/publications/dissertation/
Unfortunately, my own argument had many waves of the hand, and I didn't actually read the final product. But hopefully, it's all worked out. Alex really was a pretty careful guy, with substantial computational saavy.
Possibly, you might also enjoy this unpublished note of mine:
http://www.ucl.ac.uk/~ucahmki/jockusch.pdf
where these issues are discussed somewhat informally.
Added: I only read the comments above just now. I'm not entirely sure that the algorithm described in Perlis' thesis is any more efficient than what Pete suggests.  Let me know if it turns out to be so.