maybe they were fasting on the corresponding date to the arabic calendar....
"May be"??? so you are assuming this.
What I am telling you is fact, that 10th of Moharram, was 10th of Tevet for jews. and yes Jews fasted on that day, I don't need to check with a rabi for this. Their calender is published on internet, so 10th moharram(10th Tevet), a fasting day for Jews, no disputes there.
Only issue is, that 10th Tevet for jews, marks anniversory of a sad day. The day emperor Nebuchadnezzar laid siege to Jerusalem. hope you get my point, I am not making up any of this, nor I am assuming anything.