On the rationality of certain type A Galois representations

Abstract

Let $X$ be a complete smooth variety defined over number field $K$ and $i$ an integer. The absolute Galois group of $K$ acts on the $i$th $l$-adic etale cohomology of $X$ for all $l$, producing a system of $l$-adic representations $\{{\Phi }_l\}$. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of ${\Phi }_{\ell }$ admits a common reductive $Q$-form for all $l$ if $X$ is projective. Denote by ${\Gamma }_l$ and $G_l$ respectively the monodromy group and the algebraic monodromy group of ${\Phi }_l^{ss}$, the semisimplification of ${\Phi }_{\ell }$. Assuming that $G_{l_0}$ satisfies a group theoretic condition for some prime $l_0$ (Hypothesis A), we construct a connected quasi-split $Q$-reductive group $G_Q$ which is a common $Q$-form of $G_l^{\circ }$ for all sufficiently large $l$. Let $G_Q^{sc}$ be the universal cover of the derived group of $G_Q$. As an application, we prove that the monodromy group ${\Gamma }_{\ell }$ is big in the sense that ${\Gamma }_{\ell }^{sc}\cong G_Q^{sc}(Z_l)$ for all sufficiently large $l$.

To appear in TAMS

References in corpus (1)

• Adelic openness without the Mumford-Tate conjecture