## Local constancy in $$p$$-adic families of Galois representations.(English)Zbl 0932.32028

Let $$K$$ be a field complete with respect to a non-archimedean norm, of residue characteristic $$p$$. Let $$X$$ be a quasi-compact rigid space and $$Z$$ a closed subspace.
A substantial part of the paper deals with the question when there exists a relative tubular neighbourhood $$U$$ of $$Z$$ in $$X$$. This essentially means that $$U$$ is an admissible open containing $$Z$$ and any Galois prime to $$p$$ covering $$\mathcal E$$ of $$X$$ whose restriction to $$Z$$ is split, is also split over $$U$$, possibly after a finite extension of scalars. If $$X$$ is smooth and $$Z$$ has normal crossings, then such a relative tubular neighbourhood exists. The author therefore conjectures a rigid analogue of de Jong’s resolution of singularities [A. J. de Jong, Publ. Math., Inst. Hautes Etud. Sci. 83, 51-93 (1996; Zbl 0916.14005)], and then shows that under this hypothesis, any pair $$(X,Z)$$ admits a relative tubular neighbourhood. In particular, if the characteristic of $$K$$ is zero or if $$X$$ is the analytization of a scheme of finite type over $$K$$, then no assumption is needed. The main application of this work is the following. Let $$X$$ be a scheme of finite type over $$K$$ and let $$\mathcal L$$ be a local system which is either induced by an etale covering of $$X$$ or is a lisse sheaf of free $$\mathbf Z_l$$-modules (with $$l\neq p$$). For each rational point $$x\in X$$, let $\rho_x\: \text{Gal}(\overline K/K)\to \text{Aut}(\mathcal {L}_x)$ be the induced representation of the absolute Galois group. Then there exists a $$p$$-adic neighbourhood $$U$$ of $$x$$, such that for each rational point $$y\in U$$, we have $$\rho_x\cong\rho_y$$. In particular, if $$F$$ is discretely valued with finite residue field, then only finitely many non-isomorphic Galois representations can arise.

### MSC:

 32P05 Non-Archimedean analysis 14G20 Local ground fields in algebraic geometry

Zbl 0916.14005
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