I was curious about whether the Romans would have gone to the bother of documenting the politically embarassing issue of possible uprising in Palestine and the successful execution of a putative leader.
However, that only served to remind me of the likely fate of historical records of the time...
.....(wiki extract).....
Although there is a mythology of the burning of the Library at Alexandria, the library may have suffered several fires or acts of destruction over many years. Possible occasions for the partial or complete destruction of the Library of Alexandria include a fire set by Julius Caesar in 48 BC, an attack by Aurelian in the 270s AD, the decree of Coptic Pope Theophilus in 391, and the Muslim conquest of Egypt in 642.
After the main library was fully destroyed, ancient scholars used a "daughter library" in a temple known as the Serapeum, located in another part of the city. According to Socrates of Constantinople, Coptic Pope Theophilus destroyed the Serapeum in 391 AD.
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By way of context, Emperor Constantine's adoption of the Chi-Rho (XP) symbol was in 312 AD so we can assume that the Empire had no sense of preciousness about written records pertaining to Jesus for all of the intervening years and who knows what got damaged in the 270 AD fire?
Even if Herod had sent some type of report (of the crucifixion) to Rome then I doubt if it would have been treated much differently from the kind of corporate records discussed in an unrelated thread, the other day: - kept for half a dozen years in case of comebacks and eventually binned.