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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.

MSC:
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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