On the rationality of certain type a galois representations

Chun Yin Hui*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

Let X be a complete smooth variety defined over a number field K and let i be an integer. The absolute Galois group GalK of K acts on the ith étale cohomology group (Forumala Presented). for all primes ℓ, producing a system of ℓ-adic representations {Φ }. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φ admits a reductive Q-form that is independent of ℓ if X is projective. Denote by Γ and G respectively the monodromy group and the algebraic monodromy group of Φss,the semisimplification ofΦℓ. Assuming that Gℓ0 satisfies some group theoretic conditions for some prime ℓ0, we construct a connected quasi-split Q-reductive group GQ which is a common Q-form of Gℓ for all sufficiently large ℓ. Let GscQ be the universal cover of the derived group of GQ. As an application, we prove that the monodromy group Γ is big in the sense that (forumala Presented). for all sufficiently large ℓ.

Original languageEnglish
Pages (from-to)6771-6794
Number of pages24
JournalTransactions of the American Mathematical Society
Volume370
Issue number9
Early online date4 Apr 2018
DOIs
Publication statusPublished - Sep 2018

Keywords

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

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