Richter: You said they'd be killed publicly.
Robert Langdon: Yes, revenge. For La Purga.
Richter: La Purga?
Robert Langdon: Oh geez, you guys don't even read your own history do you? 1668, the church kidnapped four Illuminati scientists and branded each one of them on the chest with the symbol of the cross. To 'purge' them of their sins and they executed them, threw their bodies in the street as a warning to others to stop questioning church ruling on scientific matters. They radicalized them. The Purga created a darker, more violent Illuminati, one bent on... On retribution.
Contrary to what the movie states, priests can be elected Pope - they are simply elevated to the rank of Bishop before taking the office.
Ron Howard's father, Rance, makes a cameo as one of the Cardinals locked in conclave.