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Lit., "decide [all disputes] between them" - the reference being to the Torah.
i.e., the Torah.
A portion of the Book. I conceive that Allah's revelation as a whole throughout the ages is "The Book". The Law of Moses, and the Gospel of Jesus were portions of the Book. The Qur-an completes the revelation and is par excellence the Book of Allah.
The Commentators mention a particular incident when a dispute was submitted by the Jews for arbitration to the Holy Prophet. He appealed to the authority of their own books, but they tried to conceal and prevaricate. The general lesson is that the People of the Book should have been the first to welcome in Muhammad the living exponent of the Message of Allah as a whole, and some of them did so: but others turned away from guilty arrogance, relying on corrupted texts and doctrines forged out of their own fancies, though they were not conformable to reason and good sense.