ℓ-Independence for a system of motivic representations

Abstract

Let M be a motive over a number field F and v a non-archimedean valuation of F with residual characteristic p. Let ${\rho_{M,\ell} : \Gamma_{F} \rightarrow G_{M}(\mathbb{Q}_{\ell})}$ be the canonical system of ℓ-adic Galois representations associated to M, with values in the motivic Galois group G _M of M. Let ${\Phi_{v} \in \Gamma_{F}}$ be an arithmetic Frobenius element. When M belongs to a particular family of motives, we show the following (under certain hypotheses): (i) if M has good reduction at v, then for ${\ell \neq p}$, the conjugacy class of ${\rho_{M,\ell}(\Phi_{v})}$ in G _M is rational over ${\mathbb{Q}}$ and is independent of ℓ, thus giving a partial answer to a question of Serre; (ii) if M has semistable reduction at v, then the system of representations of the Weil–Deligne group ${'W_{v}}$, associated to ${\rho_{M,\ell}}$ for ${\ell \neq p}$, is a compatible system of representations of ${'W_{v}}$ with values in G _M .