Dear Tim:

How right you are to worry about computing the Jacobian of a genus one curve! This is, after all, the only non-standard step in the problem. Years ago, in the midst of many discussions with Bill McCallum, Nick Shepherd-Barron, and John Tate, I convinced myself this could be done, and it was eventually written up in the Arizona thesis of Alex Perlis:

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Possibly, you might also enjoy ths unpublished note of mine:

http://www.ucl.ac.uk/~ucahmki/jockusch.pdf

where these issues are discussed somewhat informally.

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Possibly, you might also enjoy this unpublished note of mine: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.

Dear Tim:

How right you are to worry about computing the Jacobian of a genus one curve! This is, after all, the only non-standard step in the problem. Years ago, in the midst of many discussions with Bill McCallum, Nick Shepherd-Barron, and John Tate, I convinced myself this could be done, and it was eventually written up in the Arizona thesis of Alex Perlis:

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.

Dear Tim:

How right you are to worry about computing the Jacobian of a genus one curve! This is, after all, the only non-standard step in the problem. Years ago, in the midst of many discussions with Bill McCallum, Nick Shepherd-Barron, and John Tate, I convinced myself this could be done, and it was eventually written up in the Arizona thesis of Alex Perlis:

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 ths unpublished note of mine:

http://www.ucl.ac.uk/~ucahmki/jockusch.pdf

where these issues are discussed somewhat informally.