On compatibility of the ℓ-adic realisations of an abelian motive

2017/06/28 by Johan Commelin, Commelin, Johan
Mathematics · Medicine · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cancer Treatment and Pharmacology #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1706.09444

openalex publication_date 2017/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we introduce the notion of a quasi-compatible system of Galois representations. The quasi-compatibility condition is a slight relaxation of the classical compatibility condition in the sense of Serre. The main theorem that we prove is the following: Let M be an abelian motive, in the sense of Yves André. Then the ℓ-adic realisations of M form a quasi-compatible system of Galois representations. (In fact, we actually prove something stronger. See theorem 5.1.) As an application, we deduce that the absolute rank of the ℓ-adic monodromy groups of M does not depend on ℓ. In particular, the Mumford-Tate conjecture for M does not depend on ℓ.

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