Geometric criteria for tame ramification

Abstract

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.

Néron Models and Base Change

“…We do not know how to define P C (t) intrinsically on C, without reference to an sncd-model. However, in [Ni12], the second author proved the following result.…”

“…Now assume that C is cohomologically tame and that g(C) = 1 or that δ(C) is prime to p. Note that the property that δ(C) is prime to p implies that C(K t ) is non-empty, by [Ni12, . (2.1.8) As an immediate corollary, we obtain an alternative proof of Theorem 2.1(i) in [Lo93].…”

Motivic zeta functions for degenerations of abelian varieties and Calabi-Yau varieties

“…Let A be a tamely ramified abelian K-variety. For every d in N ′ , we have This result can be seen as a particular case of a more general theory that expresses a certain motivic measure for the number of rational points on a K-variety X in terms of the Galois action on the ℓ-adic cohomology of X; see [Ni09a,Ni10d,Ni10e]. For a similar formula for the zeta function of a hypersurface singularity, see [DL02, 1.1], [NS07,9.12] and [Ni09a,9.9].…”

A logarithmic interpretation of Edixhoven's jumps for Jacobians

“…Let L be the minimal finite extension of K in K s such that C × K L has semi-stable reduction. If this extension is tame, then its degree is equal to e(C) by [Ni13,3.4.4], but this is false in general. As we have already mentioned in the introduction, it was proven by the last two authors that the stabilization index of C is equal to that of its Jacobian in the sense of (2.2.5) (see Corollary 3.1.5 in Chapter 5 of [HN14]).…”