## Divisible motives and Tate’s conjecture.(English)Zbl 1250.14005

Let $$\rho :\text{Gal}(\bar{F}/F)\rightarrow\text{GL}_{mn}(\bar{\mathbb{Q}}_l)$$ be a continuous semisimple representation, unramified almost everywhere. In a letter to Clozel in 1991, Taylor showed that $$\rho\cong\tilde{\rho}^{\oplus m}$$ for some $$n$$-dimentional $$\tilde{\rho}$$, if (a) for unramified $$v$$, the eigenvalues of $$\rho(\text{Frob}_v)$$ have multiplicity at least $$m$$, and (b) for some $$v|l$$, and some $$\tau : F_v\rightarrow \bar{\mathbb{Q}}_l$$, each Hodge-Tate number has multiplicity $$m$$. In this paper the author gives a detailed proof of this result, and extend it to motives for absolute Hodge cycles by using Tannakian duality.

### MSC:

 14C15 (Equivariant) Chow groups and rings; motives 11F80 Galois representations

### Keywords:

Galois representation; motives; absolute Hodge cycles
Full Text: