Acts 21:27 And when the seven days were almost ended, the Jews which were of Asia, when they saw him in the temple, stirred up all the people, and laid hands on him,
Who were these "Jews of Asia?"
First of all....you must understand that Jerusalem is in Asia.....Asia Minor....but nevertheless Asia. Babylon is also in Asia and a great number of Jews did not return from Babylon. Nevertheless they retained their religion and still traveled to Jerusalem for the Festivals.
In the Book of Esther is a great example of a population of Jews electing to stay behind. [Esther 2:5-7] Now in Shushan the palace there was a certain Jew, whose name was Mordecai, the son of Jair, the son of Shimei, the son of Kish, a Benjamite; Who had been carried away from Jerusalem with the captivity which had been carried away with Jeconiah king of Judah, whom Nebuchadnezzar the king of Babylon had carried away. And he brought up Hadassah, that is, Esther, his uncle's daughter: for she had neither father nor mother, and the maid was fair and beautiful; whom Mordecai, when her father and mother were dead, took for his own daughter.
Shushan is another name for Susa....the capital of Persia during the reign of Xerxes....also known as Ahasuerus. Xerxes was the Greek spelling. He was king in Persia about 424 B.C. This would have been 100 years after the return from Babylon spoken of in Ezra and Nehemiah. Persia had been part of the old Babylonia empire and Mordacai's parents, grandparents and a multitude of other Jews had elected to stay in Babylon. This is a good example of the Asian Jews spoken of in your scripture.
The Talmud....revered by many modern day Jews had its beginning in Babylon.....after the exile. You can read of some of that history right here. In [Acts 22:3] the Apostle Paul speaks of studying under the great Hebrew scholar "Gamaliel". Gamaliel's grandfather was another great Hebrew scholar called Hillel the Elder.
There were many Asian Jews in and about Jerusalem during the first century.