Someone stole the body!

[Jehanne]

Yes, the Romans pioneered federalism, and so, Judea was somewhat independent but the "sovereign power" was Rome.  And, they could crucify anyone whom the Empire deemed to be a threat to the Roman peace; as Jesus was not a Roman citizen, he did not enjoy the same rights under Roman law.  Likely, the story about him causing trouble in the Jewish temple was enough to get him arrested, and instead of a good flogging, some Jewish leaders likely convinced their Roman superiors to make an example out of him, which the Romans were all too happy to do.

Why would Pilate intervene in an internal religious matter in a potentially destabilising way? According to the bible Jesus wasn't a threat to Roman overlordship in Iudea, so therefore his heterodoxy from judaism was unlikely to be acted upon.

Thinking about the situaion I have three possibilities (assuming Jesus was real and a preacher), 1) he was the leader of a jewish messianic cult, and like later jewish messiah figures (eg bar Kochkba) was planning on overthrowing Roman overlordship and erecting a temporal theocracy centred round him (as per propesy). Thus the real Yeshua had no connection to Jesus apart from the name.
2) The gospels transferred punishment from the Sanhedrin to Pilate, because at that stage jesusism hadn't fully broken with judaism and wanted to appeal to jews still. This reading would be consistent with the bible if we accepred Jesus' divinity, miracle working and teachings.
3) Jesus was killed by neither group because he was either not real or not important enough to kill. And later christianity created a martyrdom for proselyisation purposes. Remember god figures who resurrected was a common trope in the Mediterranian world at the time, and jesusism would probably wat their god to at least match competing gods in the miracle stakes.

The Romans can and did execute any non-Roman citizens "on a whim".  What happened is that Jesus, who was born in Nazareth, was a Jewish apocalyptic preacher who "prophesied" that "the Son of Man" would come to destroy the Roman empire.  Some of his followers in Galilee encouraged him to take his message down south to Jerusalem, where he got arrested by the Romans after causing a ruckus in the Temple; after a brief examination, he was sentenced to execution, likely, with the approval of the Jewish authorities.  He and Pilate, the Govenor, probably never even met.