Ramification of surfaces: sufficient jet order for wild jumps

We previously obtained a generalization and refinement of results about the ramification theory of Artin-Schreier extensions of discretely valued fields in characteristic $p$ with perfect residue fields to the case of fields with more general valuations and residue fields. As seen in VT16, the "defect" case gives rise to many interesting complications. In this paper, we present analogous results for degree $p$ extensions of arbitrary valuation rings in mixed characteristic $(...

We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) \'etale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric \'etale fundamental group of a (geometrically connected) formal fibre of a $p$-adi...

Suppose G is a semi-direct product of the form Z/p^n \rtimes Z/m where p is prime and m is relatively prime to p. Suppose K is a local field of characteristic p > 0. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration whose dimension depends only on the ramifica...

We show that bounding ramification at infinity bounds fierce ramification. This answers positively a question of Deligne posed to the first named author.

Let $\mathcal K$ be a field of formal Laurent series with coefficients in a finite field of characteristic $p$, $\mathcal G_{<p}$ -- the maximal quotient of $\operatorname{Gal} (\mathcal K_{sep}/\mathcal K)$ of period $p$ and nilpotent class $<p$ and $\{\mathcal G_{<p}^{(v)}\}_{v\geqslant 0}$ -- its filtration by ramification subgroups in the upper numbering. Let $\mathcal G_{<p}=G(\mathcal L)$ be the identification of nilpotent Artin-Schreier theory: here $G(\mathcal L)$ is ...

This article is the second installment in a series on the Berkovich ramification locus for nonconstant rational functions f: P^1 -> P^1. Here we show the ramification locus of f is contained in a strong tubular neighborhood of finite radius around the connected hull of the critical points if and only if f is tamely ramified at all of its critical points. When the ground field has characteristic zero, this bound may be chosen to depend only on the residue characteristic. We gi...

We give a purely scheme theoretic construction of the filtration by ramification groups of the Galois group of a covering. The valuation need not be discrete but the normalizations are required to be locally of complete intersection.

We consider p-extensions of number fields such that the filtration of the Galois group by higher ramification groups is of prescribed finite length. We extend well-known properties of tame extensions to this more general setting; for instance, we show that these towers, when infinite, are ``asymptotically good'' (an explicit bound for the root discriminant is given). We study the difficult problem of bounding the relation-rank of the Galois groups in question. Results of Gord...

Let $L/K$ be an extension of complete discrete valuation fields, and assume that the residue field of $K$ is perfect and of positive characteristic. The residue field of $L$ is not assumed to be perfect. In this paper, we prove a formula for the Swan conductor of the image of a character $\chi \in H^1(K, \mathbb{Q}/\mathbb{Z})$ in $H^1(L, \mathbb{Q}/\mathbb{Z})$ for $\chi$ sufficiently ramified. Further, we define generalizations $\psi_{L/K}^{\mathrm{ab}}$ and $\psi_{L/K}^{...

We give a simple characterization of the totally wild ramified valuations in a Galois extension of fields of characteristic p. This criterion involves the valuations of Artin-Schreier cosets of the F_{p^r}^\times-translation of a single element. We apply the criterion to construct some interesting examples.