Adelic openness without the Mumford-Tate conjecture

2013/12/13 by Chun Yin Hui, Michael Larsen, Hui, Chun Yin +1 · 1 citation
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1312.3812

openalex publication_date 2013/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let X be a non-singular projective variety over a number field K, i a non-negative integer, and V_\A, the etale cohomology of X with coefficients in the ring of finite adeles \A_f over \Q. Assuming the Mumford-Tate conjecture, we formulate a conjecture (Conjecture 1.2) describing the largeness of the image of the absolute Galois group G_K in H(\A_f) under the adelic Galois representation ρ_\A: G_K -> \Aut(V_\A)=\GL_n(\A_f), where H is the Hodge group. The motivating example is a celebrated theorem of Serre, which asserts that if X is an elliptic curve without complex multiplication over K and i=1, then ρ_\A(G_K) is an open subgroup of \GL₂( \Z)⊂ \GL₂(\A_f). We state and in some cases prove a weaker conjecture which does not require Mumford-Tate but which, together with Mumford-Tate, implies Conjecture 1.2. We also relate our conjectures to Serre's conjectures on maximal motives.

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