2. The Pharisses endorsed the Roman view that he was a dangerous revolutionary, and believed that by sacrificing Jesus, they could avoid the wrath of Rome.
3. The Jewish leadership endorsed his execution.
My understanding, after 12 years of Catholic schooling, is slightly different:
1. Jesus was a Jew.
2. The Pharisses believed Jesus was a heretic, and would have had Him stoned even before the cruxifiction, but Roman rule forbade captital punishment without Roman approval.
3. Because heresy against Judaism was not a capital crime under Roman rule, the Pharisses had to convince the Romans that Jesus had commite treason against Ceaser, a capital crime.
4. Pilate was unconvinced, but believed that by sacrificing Jesus, he could avoid a Jewish uprising.