Geometric criteria for tame ramification

TL;DR

This paper establishes a formula relating tame monodromy eigenvalues to the geometry of models over discretely valued fields, with implications for semi-stable reduction, cohomological tameness, and trace formulas.

Contribution

It introduces a new A'Campo type formula for tame monodromy zeta functions and connects it to geometric criteria and trace formula error analysis.

Findings

Relates tame monodromy eigenvalues to the geometry of sncd-models.

Shows semi-stable reduction and Saito's criterion follow from the main formula.

Analyzes the trace formula error term and its implications for higher dimensions.

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 $\ell$-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.