Using the historian Josephus and Roman History - It is quite simple to prove that Jesus was still alive after Pilate left Judea in 36 AD. . All you have to do is follow the story of John the Baptist who died before Jesus started his two year mission. Remember the Baptist is killed for complaining about Herod’s second marriage to the wife of his brother Philip. To accomplish this Herod had to divorce his first wife who went crying back to her father King Aretas of Petra.
AD 26 - Pilate arrives in Judea. [confirmed]
AD 34 - Herod’s brother Philip dies. [confirmed]
AD 34 - Herod divorces wife who returns to her father King Aretas of Petra.
AD 34/35 Herod marries Philips’ wife Herodias.
AD 34/35 The Baptist complains about the marriage.
AD 34/35 Herod arrests the Baptist.
AD.35 - Lucius Vitellius becomes legate of Syria. [confirmed]
AD 35 - Herod kills the Baptist.
AD 35/36 King Aretas goes to war and wipes out Herod’s army.
AD 36 – Josephus writes the Jews thought the defeat was Gods revenge for Killing the Baptist. [confirmed]
AD 36 - Vitellius sacks Pontius Pilate who leaves Judea [confirmed]
AD 36 – Tiberius instructs Vitellius to take revenge on Aretas for Herodian war.
AD 37 – Vitllius sets out to attack.
AD 37 - Tiberius dies (16th March) [confirmed]
AD 37 - Vitellius stops attack when news arrives of Tiberius death.[confirmed]
AD 37 - Vitellius arrives back in Jerusalem to be welcomed by cheering crowds; he then cancels taxes and allows the Judean Priests custody over their own vestments, giving a period of peace [confirmed]
AD 37 - In this period of peace Jesus practices his mission.
AD 38 - Jesus is stoned to death by High Priest Theophilus ben Ananus.
So Jesus was NOT crucified by Pilate in 32 AD but died in 38 AD on a charge of blasphemy and as the Jewish Talmud says (and several other sources) he was stoned to death and then his body was hung on a tree as per the ritual.
‘If a man guilty of a capital offense is put to death and his body hung on a tree, you must not leave his body on the tree overnight. Be sure to bury him that same day, because anyone who is hung on a tree is under God’s curse. You must not desecrate the land the Lord your God is giving you as an inheritance.’ (Deuteronomy 21:22)
All the sources for this timeline can be found here - https://goo.gl/cVqiML