Divisible Motives and Tate’s Conjecture

This article gives an analog of Taylor’s trick in the context of motives for absolute Hodge cycles. The aforementioned trick, first sketched by Taylor in a letter to Clozel, gives two conditions ensuring that an mn-dimensional ℓ-adic Galois representation is divisible by m. We give a detailed proof of this result, and extend it to motives by using Tannakian duality.