Was The Torah written by Ezra the scribe?

Ezra lived sometime from around 480-440 B.C.
 
A unified, canonized Torah was available to Ezra for the public reading which took place in approximately 444 B.C.E.

Further, the various legal interpretations (mid­rashim) found in the books of Ezra and Nehemiah are themselves a result of the issues raised by a Torah in which there are apparent contradictions and repetitions. It can therefore be stated unquestionably that the canonization of the Torah was completed by the time of Ezra and Nehemiah.

The date of the return from exile, which began under Cyrus the Great, can be correlated to historical records outside the Bible that place his reign from approximately 559 to 530 B.C. The dedication of the new temple in Jerusalem, in 516 B.C., is corroborated by the records of Darius I. Certainly, there would have had to of been justification for the rebuilding of the temple through scriptural writings by the time the Temple was rebuilt.

Genesis 23:2, refers to Hebron by its pre-Israelite name Kirjath-arba.  If the Torah was written by Ezra 1,000 years later, this name would not have been known.

There is also a very interesting reference to a city that belonged to Shechem which is in the land of Canaan. Shalem, a city of Shechem, is in the land of Canaan. Now, this is found in Genesis 33:18. Well, now, Shechem was one of the most prominent cities in the 12 tribes. It was the leading city of the tribe of Ephraim. Why would the people of Egypt have to be told that Shechem was in their country if they had been living in it for six or seven centuries?

There are significant relationships to Egyptian that are to be found in the vocabulary of the Pentateuch in a unique way which you do not find in the later books or the other books of the Old Testament. In other words, there is a greater percentage of Egyptian words in the Pentateuch than is to be found in the rest of the Old Testament.

The Torah states that Moses wrote these books: Exodus 17:14; 24:4; 34:27; Numbers 33:1–2; Deuteronomy 31:9–11.

It is not a valid argument to Discount Mosaic Authorship because it was written in 3rd person.  Egyptians often wrote in 3rd person 1,500 years before Moses.  The gospel of John was certainly written in 3rd person.

We also have the witness of the rest of the Old Testament: Joshua 1:8; 8:31–32; 1 Kings 2:3; 2 Kings 14:6; 21:8; Ezra 6:18; Nehemiah 13:1; Daniel 9:11–13; Malachi 4:4.

The New Testament is also clear in its testimony: Matthew 19:8; John 5:45–47; 7:19; Acts 3:22; Romans 10:5; Mark 12:26. The divisions of the Old Testament were clearly in place in the Jewish mind long before the time of Christ, namely, the Law of Moses (first 5 books of the OT), the Prophets (the historical and prophetic books) and the Writings (the poetic books of Job, Psalms, Proverbs, etc.). So when Jesus referred to the Law of Moses, His Jewish listeners knew exactly to what He was referring.

It is not a valid argument to say that because Moses' death is at the end of Deuteronomy, he couldn't have written the book.  It is not uncommon for someone to add an obituary at the end of a book. The author of the obituary of Moses was probably Joshua, a close associate of Moses who was chosen by God to lead the people of Israel into the Promised Land (for Moses was not allowed to because of his disobedience), and who was inspired by God to write the next book in the Old Testament. A similar obituary of Joshua was added by an inspired editor to the end of Joshua's book (Joshua 24:29–33).

Conclusion: The Torah shows much Egyptian influence from around 1,500 B.C. To 1,200 B.C. The names used in several cases were those used before King Saul and not discovered later due to archeological digs. A few notations in the Torah only make sense if the Torah were written later, but there is no problem with Moses writing the bulk of the Torah and an editorial note being entered later for understanding. Ezra clearly did not write the Torah, however, he may have had a hand in copying it and compiling it in a collection along with other sacred writtings.

Resources:
http://www.jewishhistory.org/ezra-and-nehemiah/
http://www.jewishencyclopedia.com/articles/5967-ezra-the-scribe
https://en.wikipedia.org/wiki/Ezra
http://www.chabad.org/library/article_cdo/aid/111905/jewish/Ezra-the-Scribe.htm