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

Sorensen, Claus M.

Int. Math. Res. Not. 2012, No. 16, 3763-3778 (2012).

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.

Reviewer: Fumio Hazama (Hatoyama)

MSC:

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

Keywords:

Galois representation; motives; absolute Hodge cycles