Controlling the Galois images in one-dimensional families of 鈩?/span>-adic representations

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• 作者：Anna Cadoret^a ; ^{anna.cadoret@math.polytechnique.fr" class="auth_mail} ; Akio Tamagawa^b ; ^{tamagawa@kurims.kyoto-u.ac.jp" class="auth_mail}
• 关键词：primary ; 14H30 ; secondary ; 22E60 ; 22E20
• 刊名：Journal of Algebra
• 出版年：15 August 2014
• 年：2014
• 卷：412
• 期：Complete
• 页码：189-206
• 全文大小：387 K

文摘

Let k   be a finitely generated field of characteristic 0 and 鈩?  a prime. Let S   be a smooth, separated and geometrically connected scheme over k   and let 蟻:蟺₁(S)→GL_r(Z_鈩?/sub>)$蟻:{蟺}_1(S)\to {\mathrm{GL}}_r({\mathbb{Z}}_{\mathrm{鈩?/mi>}})$ be an 鈩? -adic representation of the étale fundamental group of S  . Let G   and $\overline{G}$ denote the images of 蟺₁(S)${蟺}_1(S)$ and ${蟺}_1(S_{\overline{k}})$ respectively. Given a closed point s∈S$s\in S$, we write G_s$G_s$ for the image of the absolute Galois group 螕_k(s)${螕}_{k(s)}$ of the residue field k(s)$k(s)$ viewed as a decomposition group at s   in 蟺₁(S)${蟺}_1(S)$. By previous works of the authors, it is known that, when S   is a curve, for all d≥1$d\ge 1$ and all but finitely many s∈S$s\in S$ with [k(s):k]≤d$[k(s):k]\le d$, the codimension of G_s$G_s$ in G   is at most 2. In this note, we improve this rigidity result as follows. Write g$\mathfrak{g}$, $\overline{\mathfrak{g}}$, g_s${\mathfrak{g}}_s$ for the Lie algebras of G  , $\overline{G}$, G_s$G_s$ respectively. Then for all but finitely many s∈S$s\in S$ with [k(s):k]≤d$[k(s):k]\le d$, one of the following holds: (i) the codimension of G_s$G_s$ in G   is at most 1 and g_s${\mathfrak{g}}_s$ contains $D(\overline{\mathfrak{g}}):=[\overline{\mathfrak{g}},\overline{\mathfrak{g}}]$; or (ii) the codimension of G_s$G_s$ in G   is 2 and g_s${\mathfrak{g}}_s$ contains $D^2(\overline{\mathfrak{g}}):=D(D(\overline{\mathfrak{g}}))$. We also obtain an arithmetic variant of this result, which involves the derived series of g$\mathfrak{g}$.