Because of the way the electoral college works. There will not be one vote actually cast for Barack Obama until the electors vote some 6 weeks after the general election. Until then, we are only voting for electors, not for a particular candidate.
So the ultimate authority rests in Congress, which has to certify the result of the electoral college, and can challenge any result presented based on eligibility or process concerns (note that some BDS-suffering 'rats tried to do the latter after the 2004 elections).
That said, there's no reason Congress could not delegate its authority to the FEC through a more preemptive approach. Also, the individual states have the right to set qualifications and restrictions on electors, and they could likewise enact legislation that would require that a candidate whom electors are registering as pledged for must provide proof of eligibility.
PING to post #16