Two attorneys, Richard P. Bress and Lori Alvino McGill, argue in a detailed paper published by the American Constitution Society that Congress has the authority to give the citizens of Washington, D.C., a vote in Congress.
They make two very important points. First, Congress has treated the District as a state for a number of purposes over the years. And unlike people in territories, who also have no voting representation, the residents of D.C. pay federal taxes. (Hence the motto on our license plates, "Taxation Without Representation.")
The Supreme Court weighed in on this at one point, when there was an attempt to deny residents of the District access to the federal courts under the diversity jurisdiction rule (when citizens of one state sue citizens of another) on the grounds that the District wasn't a state. Bress and McGill note some arguments expressed by Justice Rutledge in his concurring opinion in that case (which found that diversity jurisdiction applied to D.C.):
Observing that the Constitution had failed explicitly to accord District residents access to federal courts through diversity jurisdiction, Justice Rutledge remarked: "I cannot believe that the Framers intended to impose so purposeless and indefensible a discrimination, although they may have been guilty of understandable oversight in not providing explicitly against it." Having concluded that the Framers did not intend to deprive District residents of access to the federal courts, Justice Rutledge reasoned that the term "state" should include the District of Columbia where it is used with regard to "the civil rights of citizens."
Secondly, Bress and McGill argue that the founders considered the right to vote very important, and had no intent to deny it to residents of the District. The purpose of creating a separate jurisdiction for the seat of the federal government was to prevent any one state from exercising excessive control over the federal government, not to set up a jurisdiction where U.S. citizens didn't have full rights. The states had substantial power back then, and the problem of a conflict between the fledgling federal government and a state had already come up. The founders wanted to ensure that no one state could control the federal government, not deny elected representation to citizens.
Their argument is thorough and detailed. I suggest anyone interested in the subject read it. Me, I just want the at least the same representation in Congress that every other taxpaying citizen has. And don't tell me to move -- I shouldn't be deprived of representation just because I live where I work. Other citizens aren't.