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The Gemora (Gitin 57b) states that Hashem showed Dovid Hamelech a prophecy of the destruction of the two temples and he composed Tehillim Chapter 137. However the Ibn Ezra (in his introduction to his commentary on Tehillim) writes that some say it was composed at the time of the Babylonian Exile.
According to this page, the author is unknown. However, earlier in the book (fn. 2), he says that it was first printed in Livorno 5543, if that helps you.
According to Midrash Eicha Zuta, it was written by ירמיה (Jeremiah). This is also reflected in the Septuagint (aka LXX aka Targum Shivim), which opens with the line "By Jeremias, in the Captivity." This is probably the oldest tradition. According to Rashi, Midrash Tehillim (aka Midrash Secher Tov), Pesiqta Rabbatti, Ibn Ezra in his introduction to ...
In the introduction to his Moreh Nevukhei ha-Zman, Nachman Krochmal argues at length for the (Ibn Ezra's) view that this mizmor was composed at the time of the Babylonian exile.
In Yeshurin 21 Page 65 Rabbi Ezra Shaivet says that there are those who attribute it to רבי עובדיה בן דוד, however it is still unclear whether this is accurate. He does not indicate who it is that attributes this to the רבי עובדיה בן דוד.
http://he.wikipedia.org/wiki/%D7%A4%D7%A1%D7%A7%D7%99_%D7%AA%D7%95%D7%A1%D7%A4%D7%95%D7%AA Per Rabbi Yaakov Emden in his Sefer Mishna Lechem Pesachim Chapter 10 it was authored by Rabbi Yaakov ben Asher - the Tur. There are those who attribute it to the Tur's father - the Rosh. The Chida in Shem HaGedolim questions as to what the source is that it was ...
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