Question

What challenges did judaism and christianity bring to the roman empire?

Answer 1

Christianity became a tool of the Roman Empire fairly early on in it's spread. 
Religion and politics were inseparable in the ancient world, kings usually represented incarnate manifestations of their gods on earth. Polytheistic believers across the ancient Levant were accustomed to their political leaders telling them what gods were to be venerated during their rule and which deity their ruler was representative of in human form. Adding a new deity or giving a new name to an ancient deity whose belief was already established was how the conquering peoples assimilated their conquered. Tanakh recorded that any time such a practice of a Jewish king telling the Jews that they were to worship a foreign deity, the entire Jewish people suffered and did so at the very hands of the people whose deity they had left God to serve. That lesson is told right in our Jewish Bible in several dramatic narratives, the same one the Christians have as an adaptation of their Old Testament, yet they rarely see this in the story because their New Testament does not focus on the contextual meaning of the narrative, but imposes redefined meanings to support it’s dogma, often using topsy-turvy meaning to words and changes translations of phrases in a number of other places. 
Early Christian leaders did not want their flock to know the Paschal lamb represented a false man-god of Egypt, so they changed it into a sacrifice for sin to justify human sacrifice (or deicide depending on whether or not they are calling Jesus God in human form). Sin sacrifices are explained in detail in many places, and having nothing to do with the Passover sacrifice. Exodus makes no reference to the use of the Paschal lamb’s blood for expiating sin. Rather, it describes the blood on the door as an act of defiance to false gods and allegiance to the God of Israel. The sacrifice to God showed the Egyptians that the life force (blood) representing their deity was spilled by the Hebrew slaves and their god was powerless over the God of Israel to do a thing about it. It was an act of rejection of the gods of Egypt and alliance to the God of Israel, and that’s in the Torah in Exodus in context. Rather than show that Isaiah was slamming a man for calling himself a man/god representing Venus, Christian dogma personifies and makes a proper name from their Latin translation's word for star and turns that story into something about a fall of angels (no where mentioned in that narrative at ALL) to create giving of the "name" Lucifer for a demon-god of their underworld hell. Every aspect of Jewish belief is given a new spin. Hellenized Jews already apostate to Judaism after four centuries of their occupation and Roman citizens of Judea and the Galilee, desired to entice other Jews to worship as the Greeks that they believed superior in philosophy and knowledge. Jews had laws forbidding these concepts outright so they created texts that tried syncretism, their efforts to claim ,see this is what it was supposed to have been all along. However, the reality remains that those beliefs of incarnate savior deities and human sacrifice are identical to the beliefs and practices that the Torah demonized.Tammuz/Adonis (melded in Roman occupied lands along with and became Mithras worship) were incarnate sacrificed savior deities who had followers of apostate Jews in the North (Galilee) and areas of Paul's travels. Tammuz and the Romanized version of the Zoroastrian Mithras were both born of virgins (a concept having nothing to do with the Davidic Messiah or Tanakh) and their death was said to have brought their people reconciliation to their *sinful natures*. Being born with a burden of sin is a belief of the pagan peoples surrounding Judea and the Gallilee, and contradicts the Torah notion that humans may master evil inclination ( from Genesis) Tammuz was said to die and be reborn each spring. Tammuz worship had become widespread even before the destruction of the First Temple, and had so many apostate Jews as followers, it was condemned in Tanakh in the book of Ezekiel.  hope it helped :)