First read this link
"The continental shelf of a coastal State comprises the seabed and subsoil of the submarine areas that extend beyond its territorial sea throughout the natural prolongation of its land territory to the outer edge of the continental margin, or to a distance of 200 nautical miles[/B] from the baselines from which the breadth of the territorial sea is measured where the outer edge of the continental margin does not extend up to that distance"
It is worth noting, at this juncture, that the point of the Falkland Islands closest to Argentina lies some 300 miles (480 kilometres) from its coastline. "
"Article 77, points one and four address the rights of Britain to drill for oil more closely:
"The coastal State exercises over the continental shelf sovereign rights for the purpose of exploring it and exploiting its natural resources... The natural resources referred to in this Part consist of the mineral and other non-living resources of the seabed and subsoil together with living organisms belonging to sedentary species, that is to say, organisms which, at the harvestable stage, either are immobile on or under the seabed or are unable to move except in constant physical contact with the seabed or the subsoil"
Article 81 continues:
"The coastal State shall have the exclusive right to authorize and regulate drilling on the continental shelf for all purposes"
This indicates Britain is well within it's rights.
I would like to point out that the mere geographical position of an island or group of islands does not mean that the mainland country nearest has more claim than another country.
Otherwise the French would be able to claim the Channel Islands.
I would also like to point out that at first when Argentina owned the Falklands it was whilst they were still under Spanish rule. They continued to hold it for a little while after their independence until the British regained control.
It is said it was discovered by John Strong, captain of the Welfare, in January 1690.
That along with the fact that technically Argentina didn't run it at first, Spain did using Argentina as their puppets, I doesn't see how Argentina can justify their claim to these islands.
Whilst it's true that international law might not officially acknowledge British sovereignty, Argentina does not have a credible claim to the islands.