(Latecomer to this thread, and don't know quite where to make this comment, as the debate about the imperial records has flown thick and fast. So I'll just leap in here.)

In case it hasn't been pointed out yet, bear in mind that the imperial archives in Rome - and much else - were destroyed in the great fire of AD 192, as recorded separately by Galen and Dio Cassius. So it is unlikely that any census or trial records from the time of Christ would have survived to be moved to Byzantium when it became Constantinople in about AD 330.

However, Justin Martyr - writing in Rome in about AD 150 IIRC - confidently recommends the emperor to check his own imperial records for a mention of Christ's birth in Bethlehem; and also tells him to look at the Acts of Pilate for an account of the trial. There was a later (rather scurrilous) apocryphal book by the same name, but some scholars take it that, again, Justin was referring the emperor to records in the imperial archives. Christian apologists declare that Justin would not have made such bold claims if he couldn't back them up with actual bits of paper in the archives; others think he was bluffing.