The Falkland Islands have had their own Government since 1985, the oil exploration and any monies from it is down to them. Its no different from Australia being independent or New Zealand except that the Falkland Islands were uninhabited and we didn't slaughter the natives.
Its a bit ludicrous that in the 21st Century a supposedly mature democracy like Argentina, is bullying a small island community for an imagined sleight in the 19th Century that has been morphed into a huge national myth by the Peronists. Peron exploited it back in the '40s because he wanted to end the British domination of Argentine industry like the Railways. Peron himself said he didn't really believe in the claim but "it was useful to unite the people". Nothing like a spot of narrow nationalism and racial hatred to bind the people together.
The Argentine paper claim isn't worth the paper its written on, its full of outright lies and half-truths. Trouble is they say it so loud and so often that lazy journo types don't check the facts and in truth they've started to believe themselves. Well not too much, they backed right off the idea of taking it to the Hague, when someone pointed out that would mean the claim would be examined by competent judges.
See it for what it is.