## Abstract

Let X be a complete smooth variety defined over a number field K and let i be an integer. The absolute Galois group Gal_{K} of K acts on the ith étale cohomology group (Forumala Presented). for all primes ℓ, producing a system of ℓ-adic representations {Φ_{ℓ} }_{ℓ}. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φ_{ℓ} admits a reductive Q-form that is independent of ℓ if X is projective. Denote by Γ_{ℓ} and G_{ℓ} respectively the monodromy group and the algebraic monodromy group of Φ^{ss} ℓ^{,}the semisimplification of^{Φ}ℓ. Assuming that G_{ℓ0} satisfies some group theoretic conditions for some prime ℓ_{0}, we construct a connected quasi-split Q-reductive group G_{Q} which is a common Q-form of G^{◦}ℓ for all sufficiently large ℓ. Let GscQ be the universal cover of the derived group of G_{Q}. As an application, we prove that the monodromy group Γ_{ℓ} is big in the sense that (forumala Presented). for all sufficiently large ℓ.

Original language | English |
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Pages (from-to) | 6771-6794 |

Number of pages | 24 |

Journal | Transactions of the American Mathematical Society |

Volume | 370 |

Issue number | 9 |

Early online date | 4 Apr 2018 |

DOIs | |

Publication status | Published - Sept 2018 |

## Keywords

- Galois representations
- The Mumford-Tate conjecture
- Type A representations