Jesus in the Talmud
In the first few centuries CE, there were many sects of Judaism (such as Pharisees, Essenes, and Sadducees) each claiming to be the correct faith. Some scholars treat Christianity, during that era, referred to as Early Christianity, as simply one of many sects of Judaism. Some sects wrote polemics advocating their position, and occasionally disparaging rival sects. Some scholars view the depictions of Jesus in the Talmud as a manifestation of those inter-sect rivalries – thus the depictions can be read as polemics by the rabbinic authors of the Talmud which indirectly criticized the rival sect (Christianity), which was growing and becoming more dominant.
Antisemitism in Christianity
Mainstream Talmudic Rabbinical Judaism today directly descends from the Pharisees whom Jesus often criticized. During Jesus' life and at the time of his execution, the Pharisees were only one of several Jewish groups such as the Sadducees, Zealots, and Essenes who mostly died out not long after the period; indeed, Jewish scholars such as Harvey Falk and Hyam Maccoby have suggested that Jesus was himself a Pharisee. Arguments by Jesus and his disciples against the Pharisees and what he saw as their hypocrisy were most likely examples of disputes among Jews and internal to Judaism that were common at the time, see for example Hillel and Shammai.