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Geometric criteria for tame ramification

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Abstract

We prove an A’Campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field K. As a first application, we relate the orders of the tame monodromy eigenvalues on the ℓ-adic cohomology of a K-curve to the geometry of a relatively minimal sncd-model, and we show that the semi-stable reduction theorem and Saito’s criterion for cohomological tameness are immediate consequences of this result. As a second application, we compute the error term in the trace formula for smooth and proper K-varieties. We see that the validity of the trace formula would imply a partial generalization of Saito’s criterion to arbitrary dimension.

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  1. Department of Mathematics, KULeuven, Celestijnenlaan 200B, 3001, Heverlee, Belgium

    Johannes Nicaise

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  1. Johannes Nicaise
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Correspondence to Johannes Nicaise.

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The research for this paper was partially supported by ANR-06-BLAN-0183 and ANR-07-JCJC-0004.

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Nicaise, J. Geometric criteria for tame ramification. Math. Z. 273, 839–868 (2013). https://doi.org/10.1007/s00209-012-1034-8

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