Some people seem to care enormously about the circumstances of President Obama's birth. Was he perhaps born a citizen of some other country, or was he really Malcolm X's lovechild? If they could only prove some problem with his provenance, then his presidency (they imagine) would be over.
For some of these conspiracy theorists ("Birthers"), the most convincing evidence is the fact that President Obama doesn't (perhaps cannot) produce documentation sufficient to allay their suspicions. Do they imagine he's got nothing better to do than prove that he exists?
But sometimes the best strategy when faced with irrelevant accusations is to leave people wondering. For example, David Cameron refused to specify his youthful chemical indiscretions (BBC News 14 Oct 2005). A fictional example is provided by Stephen Byerley, in Isaac Asimov's short story Evidence.
Leaving something like this open to doubt (or even deliberately sowing the seeds of doubt) may serve a useful purpose. Your enemies may make a big fuss and waste a lot of energy; if they can't find something more important to talk about, people will stop paying much attention to them; meanwhile there is a mystery and glamour that may even enhance your appeal.
See my earlier post: From Malcolm X to Barack Obama.