, Volume 233, Issue 2, pp 347-363

Local constancy in families of non-abelian Galois representations

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If \(\mathcal S\) is a a scheme of finite type over a local field F, and \(\mathcal X \longrightarrow \mathcal S\) is a proper smooth family, then to each rational point \(s \in \mathcal S\) one can assign an extension of the absolute Galois group of F by the geometric fundamental group G of the fibre \(\mathcal{X}_s\) . If F has uniformiser \(\pi\) , and residue characteristic p, we show that the corresponding extension of the absolute Galois group of \(\mathcal S\) by the maximal prime to p quotient of G is locally constant in the \(\pi\) -adic topology of \(\mathcal S\) . We give a similar result in the case of non-proper families, and families over \(\pi\) -adic analytic spaces.

Received August 14, 1998