Jesus Before Pilate

What is Pilate telling the Jews?
If they aren't going to play by his rules, he isn't going to help them with Jesus.

Why do the Jews respond, “It is not lawful for us to put anyone to death” (John 18:31)?
Under Roman law, only a Roman court could sentence someone to death. That's why they had brought Jesus before Pilate in the first place.

How does their reply to Pilate fulfill what Jesus had said about "what death He would die" (John 18:32)?
Jesus had prophesied that He will die by being "lifted up" (John 3:14, see Bronze Snake), which will happen when the Roman soldiers upright the cross after nailing Him to it. The Jews' response to Pilate precluded Jesus dying by the Jewish method of execution, which was to throw stones and rocks at the victim (see Stephen stoned), who was struck down, not "lifted up." Matthew 20:18-19 records Jesus telling His disciples in even more specific terms, referring to Himself as the Son of Man: “Behold, we are going up to Jerusalem, and the Son of Man will be betrayed to the chief priests and to the scribes; and they will condemn Him to death, and deliver Him to the gentiles to mock and to scourge and to crucify."

What do the Jews tell Pilate next about Jesus?
"And they began to accuse Him, saying, “We found this fellow perverting the nation, and forbidding to pay taxes to Caesar, saying that He Himself is Christ, a King” (Luke 23:2).

Are they right?
Jesus called Himself God (see Before Abraham was, I AM), not “King.” He didn't pervert the nation, and He permitted paying taxes to Caesar: Then the Pharisees went and plotted how they might entangle Him in His talk. And they sent to Him their disciples with the Herodians, saying, “Teacher, we know that You are true and teach the way of God in truth; nor do You care about anyone, for You do not regard the person of men. Tell us, therefore, what do You think? Is it lawful to pay taxes to Caesar, or not?” But Jesus perceived their wickedness, and said, “Why do you test Me, you hypocrites? Show Me the tax money.” So they brought Him a denarius. And He said to them, “Whose image and inscription is this?” They said to Him, “Caesar’s.” And He said to them, “Render therefore to Caesar the things that are Caesar’s, and to God the things that are God’s” (Matthew 22:15-21).

Then why did the Jews lie about Jesus to Pilate?
Claiming to be king and forbidding to pay taxes to Caesar would be seditious against Rome, and therefore punishable by death on a Roman cross.