(The following story appears in the Talmud.  As Rabbi Yehoshua b. Perachia lived before the Gospel period,  there isn’t unanimous agreement that the “Yeshu” mentioned is the historical “Jesus.” Regardless, the usual interpretation of this episode is that the teacher’s unnecessary strictness drove his student, Jesus, away from learning and observing Torah).
“On his return [from political exile, Rabbi Yehoshua ben Perachia] and his students stayed at a certain inn, run by a woman, where he was shown great honor. Afterwards, he remarked, ‘Wasn’t that a nice innkeeper!’ His student, Jesus [“Yeshu” in the Talmud; “Isa” in Arabic], responded by saying, ‘But master, her eyes are crooked!’
Rabbi Yehoshua ben Perachia said, ‘Wicked person! Is this what you find important?!’ And he had him excommunicated. Each day, Jesus would come before his teacher, and ask to be forgiven. But his repentance was not accepted.
On the day that Rabbi Yehoshua finally decided to accept Jesus’ repentance, when the latter came to ask forgiveness for what he had decided would be the final time, his master was praying. When he lifted his hand to cover his eyes [to say the ‘Shema’], Jesus interpreted that gesture as yet another rejection, gave up, left, and went astray.
When later Rabbi Yehoshua said to him, ‘Return!’ Jesus responded, ‘Have you not taught us the principle that for someone who leads others astray, there is no possibility of teshuvah [repentance]?'” 
“Comment: Thus, the origin of Christianity is attributed by the Talmud to the failure of one of the greatest Sages of the Jewish People to accept the repentance of his student…” 
A further comment:
“According to Dr. [Jeffrey] Rubenstein, the account in Sanhedrin 107b recognizes the kinship between Christians and Jews, since Jesus is presented as a disciple of a prominent Rabbi. But it also reflects and speaks to an anxiety fundamental to Rabbinic Judaism. Prior to the destruction of the Temple in 70 [CE], Jews were divided into different sects, each promoting different interpretations of the law. Rabbinic Judaism domesticated and internalized conflicts over the law, while vigorously condemning any sectarianism. In other words, rabbis are encouraged to disagree and argue with one another, but these activities must be carefully contained, or else they could lead to a schism. Although this story may not present a historically accurate account of Jesus’ life, it does use a fiction about Jesus to communicate an important truth about the Rabbis. Moreover, Rubenstein sees this story as a rebuke to overly harsh Rabbis. Boyarin suggests that the Rabbis were well aware of Christian views of the Pharisees and that this story acknowledges the Christian belief that Jesus was forgiving and the Pharisees were not (see Mark 2:1–2), while emphasizing forgiveness as a necessary Rabbinic value.” 
It’s also deeply ironic that the final break occurs because of a misunderstanding. Jesus misunderstands his teacher’s covering of his eyes when saying the “Shema.” But — Wouldn’t Jesus also have covered his eyes, when saying the “Shema” himself? It would have been part of the general education of a Jewish child, let alone the practice of a talmid chacham — the student of a sage. How could he have misunderstood?
Such misunderstandings sometimes occur at moments of great emotional intensity — as when we’re approaching someone we revere to ask for forgiveness. In those moments, even casual gestures can seem laden with unintended meaning.
So, perhaps the real intent of the story is to teach us not to be unnecessarily strict or harsh towards others or — kal v’homer — towards ourselves. In that context, it’s particularly startling that the rabbis of a generation long after Jesus’ — by which time Christianity had broken from Judaism and spread widely — interpret the break, which had much to do with the question of observance, by accepting all or most of the fault for an outcome the most damaging effects of which had not yet even begun to be seen.
It’s also likely that the rabbis didn’t mean this to be understood as strictly “historical.” They might have been drawing a homiletic lesson on patience (elsewhere exemplified by Hillel) from what were, for them, “current events.” Juxtaposing Jesus with R. Yehoshua, who lived a century or more later (and whose name is the Hebrew original of the Aramaic “Yeshu”) could be a literary device to focus us on the “moral” of the story, rather than on the strict “facts” of it.
In what way did Jesus “lead others astray,” from the rabbinic point of view? He led them away from observance of the mitzvot as defined by rabbinic tradition. Here, the rabbis don’t ascribe that to some “evil” on Jesus’ part, but to a crucial error by one of their own in the way that their tradition had been transmitted.
It’s in keeping with the rabbinic principle that when misfortunes come upon us, we should first examine our own actions.
The debate about Jesus’ messiah-ship, and the divide it caused in the Jewish community, isn’t mentioned here at all.
Still, why was Rabbi Yehoshua ben Perachia specifically named? He wasn’t otherwise seen as unusually harsh in any way:
“Yehoshua ben Perachia would say: Assume for yourself a master, acquire for yourself a friend, and judge every man on the side of merit.” 
Why not make Jesus’ teacher an “anonymous” rabbi? Perhaps because Rabbi Yehoshua, living a few hundred years earlier, was already a “historical,” authoritative figure by the time that this Talmudic aggadah (anecdote) was being taught and and as such, personified the masorah (tradition).
 Sanhedrin 107b and Sotah 47a
 Rabbi Joshua ben Perahiah or Joshua ben Perachya (Hebrew: יהושע בן פרחיה) was Nasi of the Sanhedrin in the latter half of the 2nd century BCE.
 Sotah 47a
 Pirkei Avot 1:6