I've given this some more thought...his read must be, if you've got a J, or two of them, you're unlikley to bet until the turn so he must put you on the ace. Now as far I see it, a push on your part is only going to push him off an ace (he might still call but you've got to be ahead in that case) so I think the play here is another reraise. This leaves you with 20k, which while not great is not the end of the world. If he then puts you all in you've got to give him credit for the J and he may well just put you in anyway whatever his holding but I think he could also fold here.

I'm getting a lot of flack on this site (probably rightly) for wanting to avoid unecessary pushes but again I can't see the value of a push in at this stage of the tournie. With the buy in and his chip stack, I would give him credit for being a good player who would:

1. Put you on an ace.
2. Re-raise.
3. Possibly fold a hand if he knew he couldn't catch up.