If Barack was adopted by Soetoro he could have had his name legally changed to Obama as an adult.
It is just a matter of paperwork—it has no bearing on the adoption at all.
If that is the case, the trick is figuring out when and in what state.
The Senator never had a legal name change ANYWHERE in the United States.
Name changes are a matter of public record and have to be published in a publicly circulated newspaper, or so I have been informed by my local county clerk. I believe this is consistent practice throughout the nation. Though I may be positing a wrong assumption, it really makes sense to be as public as possible with something like a name change.
I have also been told, per the same county clerk, that one cannot run for public office using an alias. Using a legal name, for example the name on your birth certificate, is required for any ballots tallied for you in an election to be valid. You can run for office under an alias but then any ballots would have to be voided.