... ;it's preferable when it's based on accurate reportage though.
Such as when some people criticise the New Testament, for instance. Claims such as the documentary evidence didn't appear until 'decades after' the events, for instance. After all, documentation doesn't have to be in a written form, though in our highly literate society, we tend to assume it will.
It is amazing that you are quite content for a claim of this enormity (a guy coming back from being dead for three days) written down 20 years after the alleged event, by people you have no idea as to their integrity or motive, and with just alleged sightings and empty tombs as evidence ...
And yet if in a newspaper this morning, you read of a guy coming back from the dead in a hospital bed in London yesterday, after being dead for ten hours, you'd wonder if it was a hoax!
Strange what confirmation bias will do to a believer's mind!
You seem to be confusing what was written down 20 years after, with what survives is dated to 20 years after.
The point is what appears in what survives dated to 20 years after is a letter written to an established orthodox Christian community.
The way you present it it looks as though this established community was invented 20 years after.
Why did the roman and jewish authorities not conclusively establish a hoax at the time?
I haven't mentioned any established community so I fail to see what your main point is.
As far as the Roman and Jewish authorities not looking into a hoax is concerned, they didn't even know about this resurrection story - no one other than JCs followers knew so others couldn't read about it until at least 20 years later - so there was no need to look into any hoax story.
If the Romans had known, the history books would mention it. There are plenty of facts about the Roman occupation long before JCs time.
No one other than JC's followers knew the story of the resurrection? That is unlikely because a) It must have been preached as there is no explanation for established communities.
I think Tacitus's comment as follows put's a question mark that the story of the resurrection was not known or that there were no attempt to discredit it.
Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Christians by the populace. Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judaea, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their centre and become popular.