Controlling the Galois images in one-dimensional families of l-adic representations
Controlling the Galois images in one-dimensional families of l-adic representations
复制标题
控制 l-adic 表示的一维族中的伽罗瓦图像
DOI：
10.1016/j.jalgebra.2014.04.024
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发表时间：
2014
影响因子：
0.9
通讯作者：
Anna Cadoret and Akio Tamagawa
中科院分区：
文献类型：
--
作者：
青木さつき;入山満恵子;田中裕美子;中野泰志;S. Saito and K. Sato;Anna Cadoret and Akio Tamagawa
Let k be a finitely generated field of characteristic 0 and ℓ a prime. Let S be a smooth, separated and geometrically connected scheme over k and let ρ: π 1 (S)→ GL r (Z ℓ) be an ℓ-adic representation of the étale fundamental group of S. Let G and G¯ denote the images of π 1 (S) and π 1 (S k¯) respectively. Given a closed point s∈ S, we write G s for the image of the absolute Galois group Γ k (s) of the residue field k (s) viewed as a decomposition group at s in π 1 (S). By previous works of the authors, it is known that, when S is a curve, for all d≥ 1 and all but finitely many s∈ S with [k (s): k]≤ d, the codimension of G s in G is at most 2. In this note, we improve this rigidity result as follows. Write g, g¯, g s for the Lie algebras of G, G¯, G s respectively. Then for all but finitely many s∈ S with [k (s): k]≤ d, one of the following holds:(i) the codimension of G s in G is at most 1 and g s contains D (g¯):=[g¯, g¯]; or (ii) the codimension of G s in G is 2 and g s contains D 2 (g¯):= D (D (g¯)). We also obtain an arithmetic variant of this result, which involves the derived series of g.
