None of the books in the Bible were written by the alleged Apostles. Paul only claims to have met the Apostles Peter and James (Gal 1:19). Later in Galatians (2:9) he also mentions a John, so whoever this John was it was not an Apostle. Incidentally he never claims to have met Jesus. No writings by Peter, James, or John survive. So apparently some time after 50 AD Paul met some people who were involved in the Jerusalem Church and had a big theological argument with Peter. We don't know what they claimed about anything and we don't know how any of them died. All we can surmise is that Peter believed in circumcision and Paul didn't (assuming Paul is being honest about the disagreement), and that Paul thought that James was "the lord's" brother (presumably Jesus). BTW, we don't even know if this James was a blood relative because the word brother could refer to any man who was a fellow Christian.
I should at this point mention Josephus, because he mentions someone named James, the brother of Jesus who was called Christ. This passage is used by Christians as evidence for both the biblical James and Jesus. However, the actual quote will demonstrate that this is not the case. The passage occurs In The Antiquities of the Jews Book XX Chapter 9.
The Antiquities of the Jews/Book XX - Wikisource, the free online library
"1. AND now Caesar, upon hearing the death of Festus, sent Albinus into Judea, as procurator. But the king deprived Joseph of the high priesthood, and bestowed the succession to that dignity on the son of Ananus, who was also himself called Ananus. Now the report goes that this eldest Ananus proved a most fortunate man; for he had five sons who had all performed the office of a high priest to God, and who had himself enjoyed that dignity a long time formerly, which had never happened to any other of our high priests. But this younger Ananus, who, as we have told you already, took the high priesthood, was a bold man in his temper, and very insolent; he was also of the sect of the Sadducees,
[23] who are very rigid in judging offenders, above all the rest of the Jews, as we have already observed; when, therefore, Ananus was of this disposition, he thought he had now a proper opportunity [to exercise his authority]. Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrim of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others, [or, some of his companions]; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned: but as for those who seemed the most equitable of the citizens, and such as were the most uneasy at the breach of the laws, they disliked what was done; they also sent to the king [Agrippa], desiring him to send to Ananus that he should act so no more, for that what he had already done was not to be justified; nay, some of them went also to meet Albinus, as he was upon his journey from Alexandria, and informed him that it was not lawful for Ananus to assemble a sanhedrim without his consent.
[24] Whereupon Albinus complied with what they said, and wrote in anger to Ananus, and threatened that he would bring him to punishment for what he had done; on which king Agrippa took the high priesthood from him, when he had ruled but three months, and made Jesus, the son of Damneus, high priest."
So as we can see Ananus had James killed on trumped up charges. King Agrippa punished Ananus for what he did by taking away his high priesthood and giving it to James' brother Jesus. Jesus and James are sons of Damneus, not Joseph/Mary. Jesus survived James which is the opposite of biblical James and Jesus. Furthermore the word christ means "anointed" (more specifically anointed with oil). Who gets anointed with oil? Kings and high priests. When Christians point to this passage as evidence for Jesus they either never read it themselves, or hope that you don't.
Pilate never wrote about Jesus.
Because of contemporary accounts of him.
