## 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 |

### Funding

Received by the editors May 3, 2016, and, in revised form, January 9, 2017, and January 10, 2017. 2010 Mathematics Subject Classification. Primary 11F80, 14F20, 20G30. Key words and phrases. Galois representations, the Mumford-Tate conjecture, type A representations. The present project was supported by the National Research Fund, Luxembourg, and cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND).

Funders | Funder number |
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Marie Curie | |

European Commission | FP7-COFUND |

Fonds National de la Recherche Luxembourg | |

Seventh Framework Programme |

## Keywords

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