Local constancy in \(p\)-adic families of Galois representations.

*(English)*Zbl 0932.32028Let \(K\) 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.

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.

Reviewer: Hans Schoutens (Middletown)