Mathematische Zeitschrift

, Volume 233, Issue 2, pp 347–363

Local constancy in families of non-abelian Galois representations

  • Mark Kisin
Original article

DOI: 10.1007/PL00004802

Cite this article as:
Kisin, M. Math Z (2000) 233: 347. doi:10.1007/PL00004802


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.

Mathematics Subject Classification (1991): 32P05, 14G20.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Mark Kisin
    • 1
  1. 1.Department of Mathematics and Statistics, University of Sydney, Sydney 2006, Australia AU