To start, I will repeat what my Jewish Studies professor taught me about the origins of Christianity and Judaism. His claim was that because of the destruction of the Second Temple in 70 CE, rabbinic Judaism as an established religion did not begin till the first century. Therefore, Christianity (which began to develop after Jesus' resurrection in 30 CE-33 CE) and Judaism as institutional religions began to develop around the same time.

When I studied Kabbalah (Jewish Mysticism) with this same professor he made another observation about its development in relation to Christianity. Hasidism more specifically has close ties to Christian thought in its early development. In the 18th Century Hasidism developed as a mystical revival of Judaism, and began to gain immense popularity especially among commoners. Up until that point, layman's access to religious spiritualism within Judaism was extremely limited. Hasidism began to espouse the idea that dedication to a religious leader or rebbe was integral to connection with God. Additionally, God's presence on Earth in Creation gave access to experiencing Him. Intentional worship and internal spirituality also began to play a larger role in this movement. Dedication to following the law and strict scholarship on scriptures began to take a back seat.

To be clear, the growth of Hasidism cannot be attributed to Christianity. However, the path towards a more "incarnational", accessible (in ways similar to Christianity) and less scholarly (Rabbinic) tradition is indicative of the cross-cultural interactions between Judaism and Christianity. The rebbes developing Hasidism were aware of Christian theological tradition and how their traditions were beginning to share general ideas.

Sources: My professor, and Arthur Green's review of Shal Magid's Hasidism Incarnate: Hasidism, Christianity, and the Construction of Modern Judaism

https://ejournals.bc.edu/ojs/index.php/scjr/article/view/9120

I think this question is a little big. One can come from it from a lot of different ways. Jews and Christians have had relations for about two thousand years, so where and when ought one talk about influences? Jews certainly were influenced by facets of Christianity through the centuries. Sefer Hasidim (a mystical text from 12th Century CE Rhineland) remarks, "As the Gentiles do, so do the Jews."

Anyway, Rabbinic Judaism emerges as a continuation of Pharisaic Judaism. Pharisaic Judaism was one of the Jewish movements active in Judea during the Second Temple period (the others being, according to Josephus, the Sadducees, the Essenes, and the Zealots). The Pharisees believed in following the Torah according to the traditions of their sages, and the Pharisaic teachers were popular among the common people. For a variety of reasons, Pharisaic leaders were in a prime position at the fall of Jerusalem in 70 CE to serve as leadership.

The Pharisaic/Rabbinic leaders following the fall of Jerusalem reached a simple conclusion: this catastrophe had occurred due to senseless hatred and divisiveness. Rabban Gamaliel II served as the president of the Jewish Court and the head of the Jewish academy in Yavneh shortly after the destruction of Jerusalem (within a decade or so). Gamaliel II pursued a policy of using proper jurisprudence to work through major topics of debate. By using a method of proper jurisprudence, a consensus could be reached which all Jewish scholars would be obliged to accept. In doing this, the Jewish community could become more cohesive, accepting, and united. However, rejection of proper jurisprudence entailed entertaining thoughts of sectarianism; the same thoughts which had led to the destruction of Judea.

We see these themes at play in the famous story of the Oven of Akhnai (BT Bava Metzia 59b). The Jewish sages are debating an issue, and they all are in agreement except for one. Rabbi Eliezer ben Hyrcanus rejects the consensus made by the other scholars, and as evidence he produces a number of miracles. Yet miracles are not evidence in legal debates. The position of the Jewish scholars is that "[the Torah] is not in heaven" (Deut. 30:12), it was given to humanity to be understood using reason and a democratic process. Rabbi Eliezer ben Hyrcanus is not following proper jurisprudence and refuses to recognize a legitimately reached decision made by the community. By doing so, he is jeopardizing the safety of the Jewish people. Although all the scholars respect Rabbi Eliezer ben Hyrcanus, they have no choice but to excommunicate him (not for having a different view, but for not respecting the legal process and the unity of the community). When Eliezer is excommunicated, Gamaliel II was on a boat. A great storm begins to rage, and Gamaliel II knows that this storm was caused due to his decision to excommunicate Eliezer. Gamaliel II cries out:

Master of the Universe, it is revealed and known before You that neither was it for my honor [that] I acted [when ostracizing him], nor was it for the honor of the house of my father that I acted; rather, [it was] for Your honor, so that disputes will not proliferate in Israel.

With this, the storm calms. Rabban Gamaliel II was heavy-handed, but he did pursue an important goal. He knew that for the Jewish people to survive, they needed to focus on building a united community with a respected legal process.

This is relevant in regards to Christianity because Jewish-Christians were a sectarian group. They did not accept proper jurisprudence. Jewish-Christians were not the only sectarian group, but they quickly became the most prominent sect following the destruction of Jerusalem. The Jewish concern over divisiveness was not caused because of Christianity, however Jewish-Christians by their very nature were sectarians and to be understood as such.

In a related issue, Christianity offered a different understanding of divine law that Rabbinic Judaism (the Oven of Akhnai story plays with this too). As the Greeks understood it, divine law was the unwritten, universal, natural law as revealed by reason (logos). Positive law as made by humans was at best a written, particular construct meant to point toward the divine law. Judaism regards the Torah as divine law, instead, because it was commanded by the Divine. However, from a Greek point of view, the Torah does not look like divine law (it is written, particular, arbitrary, etc.). So what is a Jew to do when he encounters Greek understandings of law? Some Hellenistic Jews attempted to see in the Torah the essence of divine law, such as Philo. Christine Hayes writes (What's so Divine about Divine Law?, chapter 3):

It is Philo of Alexandria (fl. first century CE) who goes the furthest in resolving the cognitive dissonance between Greco-Roman and biblical conceptions of divine law. He asserts the identity of the two and then labors to demonstrate that the Torah of Moses possesses all of the properties and qualities of Greek natural law: it is self-identical with universal truth, which entails its rationality, its immutability, and its unwritten character.

So Philo argues that the Torah is divine law, it's simply the case that one has to dig a little deeper in order to see this to be the case. The Christian apostle Paul takes a uniquely different approach. This can be seen evidently in Galatians (3:23-28):

Now before faith came [with Jesus], we were imprisoned and guarded under the law [of the Torah] until faith would be revealed. Therefore the law was our disciplinarian [pedagogue] until Christ came, so that we might be justified by faith. But now that faith has come, we are no longer subject to a disciplinarian, for in Christ Jesus you are all children of God through faith. As many of you as were baptized into Christ have clothed yourselves with Christ. There is no longer Jew or Greek, there is no longer slave or free, there is no longer male and female; for all of you are one in Christ Jesus.

So for Paul, the Torah was an interesting example of a positive law given by God. It pointed toward the natural, divine law. This natural, divine law was revealed and established with the coming of Jesus (who is equated with the Logos in Christian literature). By having fidelity to Jesus, one is able to follow the divine law and has no use for the positive law of the Torah. Hayes writes (What's Divine about Divine Law?, chapter 4):

The metaphor of the Law as pedagogue guarding Israel in the interim period between the promises to Abraham and the arrival of Christ echoes the Platonic depiction of human positive law as a substitute for the direct rule of the gods or of sages able to perceive eternal verities (G-R 3(i) and (ii)). Plato understood conventional law to be a second-best accommodation to our imperfect natures. It has the effect of curbing sinful behavior, but it is inadequate for the cultivation of human virtue, for which a savior (soter, soterion) is required, in the form of a person or persons of discernment whose commitment to the logos qualifies them to guide others to virtue (G-R 6).

Further, she writes:

Pedagogues were often ridiculed for being harsh disciplinarians who employed verbal reproach and physical punishment (ibid.). Philo defended the harsh discipline of the pedagogue because of the latter’s good intentions: the pedagogue sought to improve the souls of his charges (Philo, Migrations 116, as cited in Lull, 490). As a moral disciplinarian, the pedagogue was associated with “teaching self-restraint” and belonged therefore to the institutions of “education toward virtue” (Lull, 491). Inculcating self-restraint conformed to Platonic notions of education according to which nurses, parents, and pedagogues should keep childish excesses in check (Plato’s Laws 808d–e), while teaching children—with threats and beatings if necessary—to excel in doing what is just, noble, and holy, and to refrain from doing the opposite (Plato’s Protagoras 324d–325d; Laws 653b–c; see Lull, 491). This notion of education considers children to be little different from slaves to the extent that they, like slaves, were believed to be governed by “unbridled passions” (Lull, 494). Children are depicted as resentful of their pedagogue’s constraints and happy to exchange their pedagogues “for reason as the divine guide of their lives when they passed from immaturity to maturity” (ibid.).

This context helps to further explain Paul's overall approach to the Torah and also to Jesus.

Continued