I don't believe there is a trump card in the sense that with relatively small investment you can destroy an american carrier. I've said it before, there's no silver bullet, no one huge weakness that's readily open to exploit.
There's two main ways to go about it. Either take a high tech, expensive route, also route that is uncertain and that might take decades, where the goal is to have technology superior than what the carrier has prepared for. Or you use whatever level of lower tech systems in accordance with huge numbers to overwhelm the defenses. Such approach takes less time to prepare and is more certain. It can again be approached from two sides - less human losses to achieve the goal and much higher money cost, or big numbers of humans for a relatively small amount of money. Humans would there play the role of computer/navigation/sensor/targeting/etc. It all depends on what you can, as a country that is attacking a us carrier, afford yourself.
Locating and targeting goes the same way. it can be done with relatively low tech systems too, you just have to have a damn big number of them and be prepared for big losses.
Something else now, seemingly off topic but not really, i'll explain later. I've tried to get an answer on defencetalk forum but no one did answer. anyway. Is there a two-way datalink system that can't be properly jammed if the transmitter and reciever are close enough? I'm not talking just radio waves, any kind of data link, from laser to soundwaves to physical shockwaves to whatever you can think of. Also, what would be that distance where the link cant be jammed in todays war enviroment for your proposed system? Any kind of actual physical link is out of the question, though.