I dont know how true this is but found this on a web site it says we actually share the resources with Argentina in an agreement reached in 1995.
One of the first acts of Carlos Menem, on becoming president of Argentina in 1989, is to open peace negotiations with Britain. They make little immediate progress, but hostilities are formally concluded by 1995.
A treaty in 1995 also tackles another important issue which has been a subtext in the conflict. Geological surveys suggest that there may be extensive oil reserves in the Falklands region. This, as much as national pride, is a reason for claiming possession.
Although no progress is made on the matter of sovereignty, a compromise on oil is reached in 1995. Britain and Argentina agree to share, in proportions varying in different regions, any wealth deriving from the anticipated oil fields. The British share is to be used to defray the cost of the war and the garrison. Licences are issued in 1996. Exploratory drilling begins in 1998.
Read more:
http://www.historyworld.net/wrldhis/plaintexthistories.asp?historyid=ac51#ixzz344bCBS5P