Ten points about the Tenth of Teves from an article by Rabbi Berel Wein.
1) The Tenth of Tevet marks the onset of the siege of Jerusalem by Nebuchadnezzar, the King of Babylonia, and the beginning of the battle that ultimately destroyed Jerusalem.
2) The date of the Tenth of Tevet is recorded for us by the prophet Yechezkel, who himself was already in Babylonia as part of the first group of Jews exiled there by Nebuchadnezzar, 11 years earlier than the actual destruction of the Temple in Jerusalem itself.
3) The Tenth of Tevet is viewed as such a severe and important fast day that it is observed even if it falls on a Friday (erev Shabbat), while our other fast days are so arranged by calendar adjustments as to never fall on a Friday, so as not to interfere with Shabbat preparations.
4) On the eighth of Tevet, King Ptolemy of Egypt forced 70 Jewish scholars to gather and translate the Hebrew Bible into Greek. Even though the Talmud relates to us that this project was blessed with a miracle
5) The 70 scholars were all placed in separate cubicles and yet they all came up with the same translation
6) The general view of the rabbis of the time towards this project was decidedly negative. The Talmud records that when this translation became public “darkness descended on the world.”
7) The ninth day of Tevet is held to be the day of the death of Ezra the Scribe. This great Jew is comparable even to Moses in the eyes of the Talmud. “If the Torah had not been granted through Moses, it could have been granted to Israel through Ezra.”
8) Ezra led the return of the Jews to Jerusalem from their Babylonian exile. It was under his direction and inspiration, together with the help of the court Jew, Nechemiah, that the Second Temple was built, albeit originally in a much more modest scale and style than the grandeur of Solomon’s Temple.
9) Since fasting on the eighth, ninth and 10th days of Tevet consecutively would be unreasonable, the events of the eighth and ninth were subsumed into the fast day of the Tenth of Tevet.
10) The rabbinic policy of minimizing days of tragic remembrances played a role in assigning the Holocaust remembrance to the Tenth of Tevet for a large section of the Israeli population.