Though Roman Catholic, I am of Hebrew ethnicity. With family and friends who are members of Judaism (as well as a very close friend who is a rabbi) I hope I can help answer your question.
First, there is no “official” Jewish position on this matter. Unlike Catholicism and other Christian denominations, there is no central authority in Judaism. While some Jews still hold a belief in a coming personal Messiah, others do not. Some Jews who believe in the Messiah do not hold a belief in a personal one, (i.e., some may view the Messiah as an ideology more than a person). Therefore it is not a “core tenant” of Judaism as a whole.
Being Jewish is less about doctrine (and therefore not about whether one believes in the coming of a personal Messiah). It’s more about heritage to be exact. For example (and don’t hold your breath while I explain this because you will pass out):
Being Jewish is being part of what Jews call “the Tribe.” Most often it means being one who practices some form of Judaism. But there are secular Jews who do not practice Judaism, some of whom are agnostic and some who are even atheist. Then, like the apostles, there are Jews who believe that Jesus (Yeshua) of Nazareth is the promised Messiah (I am one of them). Others believe in a different person or version of the Messiah (some believe the current State of Israel is the “fulfillment” of the promises traditionally associated with the Messiah). Some are converts, most are members by DNA.
Some people don’t consider me a Jew, while others, like my rabbi friend, pronounced me a Halachic Jew before I even knew what that was (my parents are children of Jews, who are children of Jews, of Jews, etc.). Others consider only those who are Orthodox as Jews. And then there are the Sephardic (which includes the Mizrahi) and the Ashkenazi divisions which among them you will meet some who may or may not consider one or the other as truly Jewish.
As the old joke goes, if left to themselves, two Jews will build three synagogues for worship: one for the first to worship in, the second for the other, and the third will be the synagogue neither would be caught dead in.
As to Scriptures that the Messiah is to fulfill and what is expected of him, if you’re expecting an official list here too, you need to give that up. You also need to think outside the “Old Testament” as not all the teachings about the Messiah are found there.
But among the Old Testament/Hebrew Scriptures one will find the same texts the Christians use among Jews who expect a personal Messiah. The only difference is that this usually involves a fulfillment with his first (and only) coming (there are no two advents of the Messiah in Judaism). What a Christian expects to happen at the Second Coming is more or less what Jews expect(ed) when the Messiah appears (or appeared–and this is why many did not accept Yeshua as Messiah).
You also need to understand that some of the concepts of the Jewish beliefs about the Messiah are found in the Talmud. You can’t actually have a good understanding of what Jews believe or why they reject what they do about the Messiah concept without learning the Talmudic view.
I’ve tried to offer this before to other Christians and fellow Catholics who just got upset with it. Even when I tried to go point by point they got mad and often said the same thing: “Why are you confusing the point? I want to know according to my Christian understanding what they understand not what they understand from their point of view!” But it doesn’t work that way. When it comes to Judaism, you have to accept it on its own terms and not try to make it fit into some other ideology…not even a Christian one.
A very limited overview that might be helpful is provided on Wikipedia under
Jewish Messianism.