A uniform bound for the order of monodromy

For a compatible system of $\ell$-adic Galois representations with some condition, the elements of inertia acting unipotently constitutes an open subgroup. We give an explicit bound for the index of this subgroup, which is independent of $\ell$. For the etale cohomology of a smooth projective variety, we give a bound of the index acting unipotently on the all degree of cohomology groups. This bound depends only on some numerical invariants of the variety such as Betti numbers and this is independent of $\ell$.

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Accepted 22 October 2015

Published 8 July 2016