In relation to Obama? With what is legally known, what can be done according to the Constitution?
Lots of things. For starters, Congress could have demanded his credentials when they counted the electoral votes. Not even a single Republican demanded that the Constitution's provisions be followed in this respect.
Dozens of judges have had the opportunity to subpoena the documents. They've all refused.
A congressional committee could do the same. But the Republicans are way too cowardly to do that.
None of them care one whit about their oath, or what the Constitution requires of them.
The twentieth amendment, section three REQUIRES that Congress ensure the eligibility status of a President elect and if he/she is ineligible to serve, must name a replacement.
The burden of proving eligibility is on the President elect . This means that there is no way that someone who cannot prove eligibility can LEGALLY serve as President. If Congress does not have the answer as to whether or not Obama is "legal" or not, then the mere absence of such evidence is proof that he is in fact illegal.
What can be done? My opinion is that we are still not under the beginning of a Presidential term in office because such a term can only begin with a legal President. This means that section three is still in play and anyone under the oath of office from Article six can demand to see documentation that Obama meets the eligibility requirements for President.