This is kind of hilarious. Uruguay is like the only place to NOT have a claim to the islands.
Look it seems like it should be Argentinas, but the UK has been the administration there since the 30's, and the best Argentina can do is say they "inherited" it from Spain in the early 1800's. Spain and the UK both claimed it.....and left. It was uninhabited when Europeans went there a few hundred years ago. The 1982 invasion was from a ridiculous leader in Argentina who was trying to deflect from the economic condition of the country.
If its one thing Argentina has (besides a beautiful country), its national pride in spades. So the "war" did its job, just under a thousand people died, and it still belongs to the UK. The biggest point of all? The resident WANT it to remain British.
So here is what should happen. Argentina and UK find a way to share in investment and distribution of the resouces, and Britain doesn't expand drilling into Argentinas maritime claims locally out of respect.
Don't the sheep here outnumber the people 200 to 1?