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Local constancy in families of non-abelian Galois representation. (English) Zbl 0980.14019
From the introduction: Suppose \(F\) is a field complete with respect to a non-archimedean valuation, with uniformiser \(\pi\) and residue characteristic \(p>0\), and \({\mathcal S}\) a scheme of finite type over \(F\). If \(\rho\) is a representation of \(\pi_1({\mathcal S})\), the étale fundamental group of \({\mathcal S}\), then \(\rho\) induces for each rational point \(s\in{\mathcal S}\) a representation \(\rho_s\) of the absolute Galois group of \(F\). In our previous work [M. Kisin, Math. Z. 230, No. 3, 569-593 (1999; Zbl 0932.32028)], we showed that if \(\rho\) is a representation on a finite free \(\mathbb{Z}/n\) module (for any \(n\in\mathbb{N}^+)\) or a finite free \(\mathbb{Z}_\ell\) module for a prime \(\ell\) different from \(p\) then the representations \(\rho_s\) are locally constant in the \(\pi\)-adic topology of \({\mathcal S}\). In the present paper we prove results which are non-abelian analogues of the results of the paper cited above. More precisely we prove:
Theorem. Let \({\mathcal S}\) be a connected \(F\)-scheme of finite type, \(f:{\mathcal X}\to{\mathcal S}\), proper and smooth, and \({\mathcal Z}\subset {\mathcal X}\) a divisor with normal crossings relative to \({\mathcal S}\). Put \({\mathcal U}= {\mathcal X}-{\mathcal Z}\). We assume that the fibres of \(f\) are geometrically connected. If \(s\in S\) is a closed point, the prime to \(p\) geometric étale fundamental group \(\pi^{(p)}_{1,\text{geom}} ({\mathcal U}_s)\) does not depend on the choice of \(s\), and for each rational point \(s\in {\mathcal S}\) we denote by \(E_s\) the corresponding extension of \(\text{Gal} (\overline F/F)\) by \(\pi_1^{ (p)} ({\mathcal U}_s)\). The extensions \(E_s\) are locally constant in the \(\pi\)-adic topology of \({\mathcal S}\). Namely for any rational point \(s\in{\mathcal S}\) there is a \(\pi\)-adic neighbourhood \({\mathcal V}\subset {\mathcal S}\) containing \(s\) such that \(E_s @>\sim>> E_t\) (as extensions) for every rational point \(t\in {\mathcal U}\).
The proof uses rigid analytic methods.
A consequence of the above theorem is that if \(F\) has discrete valuation and finite residue field, and \({\mathcal S}\) has a system of coordinates (i.e. a global embedding \({\mathcal S} \hookrightarrow \mathbb{A}^n)\), then the rational points of \({\mathcal S}\) where these coordinates have \(\pi\)-adic norm \(\leq 1\) give rise to only a finite number of isomorphism classes of extensions.

14G20 Local ground fields in algebraic geometry
14F35 Homotopy theory and fundamental groups in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
11S20 Galois theory
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