We compute the higher ramification groups and the Artin conductors of radical extensions of the rationals. As an application, we give formulas for their discriminant (using the conductor-discriminant formula). The interest in such number fields is due to the fact that they are among the simplest non-abelian extensions of the rationals (and so not classified by Class Field Theory). We show that this extensions have non integer jumps in the superior ramification groups, contrarily to the case of abelian extensions (as prescribed by Hasse-Arf theorem).
The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin-Schreier extensions. We define refined versions of invariants of ramification in the classical ramification theory and compare them. Furthermore, we can treat the defect case.
For a smooth surface X over an algebraically closed field of positive characteristic, we consider the ramification of an Artin-Schreier extension of X. A ramification at a point of codimension 1 of X is understood by the Swan conductor. A ramification at a closed point of X is understood by the invariant r_x defined by Kato [2]. The main theme of this paper is to give a simple formula to compute r_x' defined in [4], which is equal to r_x for good Artin-Schreier extension. We also prove Kato's conjecture for upper bound of r_x.
For a rational prime $p\neq 2$, we compute the sequence of ramification groups of a Galois, radical and finite extension $L/F$ where $F/\mathbb{Q}_p$ is an unramified finite extension. First, we compute it in the case where the exponent of Gal$(L/F)$ is a power of $p$, by a Hecke's theorem and some inductions,. Finally, we deal with the general case. For the case $p=2$, the same method can work, provided that we make an additional hypothesis.
Let $k$ be a complete discretely valued field of equal characteristic $p > 0$ with possibly imperfect residue field and let $G_k$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on $G_k$ coincide with the differential Artin conductors and Swan conductors of Galois representations of $G_k$. As a consequence, we give a Hasse-Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse-Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors a…
In local class field theory, the Schmid-Witt symbol encodes interesting data about the ramification theory of $p$-extensi-ons of $K$ and can, for example, be used to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula for the Schmid-Witt symbol of Artin-Schreier extensions of local fields. Later, his formula was generalized to Artin-Schreier-Witt extensions, but still over a local field. In this paper we generalize Schmid's formula to compute the Artin-Schreier-Witt-Parshin symbol for Artin-Schreier-Witt extensions of two-dimensional local …
Let $K$ be a local field of characteristic $p$ and let $L/K$ be a totally ramified elementary abelian $p$-extension with a single ramification break $b$. Byott and Elder defined the refined ramification breaks of $L/K$, an extension of the usual ramification data. In this paper we give an alternative definition for the refined ramification breaks, and we use Artin-Schreier theory to compute both versions of the breaks in some special cases.
Let $π: Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship between the $a$-numbers of $Y$ and $X$ and the ramification of the map $π$. This is analogous to the relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$ and the ramification of the cover, so we instead give bounds on the …
We study questions of multiplicities of discriminants for degenerations coming from projective duality over discrete valuation rings. The main result is a type of discriminant-different formula in the sense of classical algebraic number theory, and we relate it to Artin conductors via Bloch's conjecture. In the case of discriminants of planar curves we can calculate the different precisely. In general these multiplicities encode topological invariants of the singular fibers and in the case of characteristic $p$, wild ramification data in the form of Swan conductors.
We classify stacky curves in characteristic $p > 0$ with cyclic stabilizers of order $p$ using higher ramification data. This approach replaces the local root stack structure of a tame stacky curve, similar to the local structure of a complex orbifold curve, with a more sensitive structure called an Artin-Schreier root stack, allowing us to incorporate this ramification data directly into the stack. As an application, we compute dimensions of Riemann-Roch spaces for some examples of stacky curves in positive characteristic and suggest a program for computing spaces of modular forms in this set…
Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field $k$ of characteristic $p>0$ and $B \subset X(k)$ is a finite, possibly empty, set of points. The Newton polygon of a degree $p$ Galois cover of $X$ with branch locus $B$ depends on the ramification invariants of the cover. When $X$ is ordinary, for every possible set of branch points and ramification invariants, we prove that there exists such a cover whose Newton polygon is minimal or close to minimal.
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 $(0,p)$ in a more general setting. More specifically, the only assumption here is that the base field $K$ is henselian. In…
Let k be a complete discrete valuation field of equal characteristic p>0. Using the tools of p-adic differential modules, we define refined Artin and Swan conductors for a representation of the absolute Galois group $G_k$ with finite local monodromy; this leads to a description of the subquotients of the ramification filtration on $G_k$. We prove that our definition of the refined Swan conductors coincide with that is given by Saito, which uses etale cohomology. We also study its relation with the toroidal variation of the Swan conductors.
For a local field K with positive residue characteristic p, we introduce, in the first part of this paper, a refinement bAr_K of the classical Artin distribution Ar_K. It takes values in cyclotomic extensions of Q which are unramified at p, and it bisects Ar_K in the sense that Ar_K is equal to the sum of bAr_K and its conjugate distribution. Compared with 1/2 Ar_K, the bisection bAr_K provides a higher resolution on the level of tame ramification. In the second part of this article, we prove that the base change conductor c(T) of an analytic K-torus T is equal to the value of bAr_{K} on the Q…
We define generalizations of classical invariants of wild ramification for coverings on a variety of arbitrary dimension over a local field. For an l-adic sheaf, we define its Swan class as a 0-cycle class supported on the wild ramification locus. We prove a formula of Riemann-Roch type for the Swan conductor of cohomology together with its relative version, assuming that the local field is of mixed characteristic. We also prove the integrality of the Swan class for curves over a local field as a generalization of the Hasse-Arf theorem. We derive a proof of a conjecture of Serre on the Artin c…
We give necessary and sufficient conditions for the Kummer extension $K:=\mathbb{Q}\left(ζ_n,\sqrt[n]α\right)$ to be monogenic over $\mathbb{Q}(ζ_n)$ with $\sqrt[n]α$ as a generator, i.e., for $\mathcal{O}_K=\mathbb{Z}\left[ζ_n\right]\left[\sqrt[n]α\right]$. We generalize these ideas to radical extensions of an arbitrary number field $L$ and provide necessary and sufficient conditions for $\sqrt[n]α$ to generate a power $\mathcal{O}_L$-basis for $\mathcal{O}_{L\left(\sqrt[n]α\right)}$. We also give sufficient conditions for $K$ to be non-monogenic over $\mathbb{Q}$ and establish a general crit…
Let K and F be complete discrete valuation fields of residue characteristic p>0. Let m be a positive integer no more than their absolute ramification indices. Let s and t be their uniformizers. Let L/K and E/F be finite extensions such that the modulo s^m of the extension O_L/O_K and modulo t^m of O_E/O_F are isomorphic. Let j=<m be a positive rational number. In this paper, we prove that the ramification of L/K is bounded by j if and only if the ramification of E/F is bounded by j. As an application, we prove that the categories of finite separable extensions of K and F whose ramifications ar…
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 the group obtained from a suitable profinite Lie $\mathbb{F}_p$-algebra $\mathcal L$ via the Campbell-Hausdorff composit…
Suppose $K$ is a finite extension of $\mathbb{Q}_p$ containing a $p^M$-th primitive root of unity. For $1\leqslant s<p$ denote by $K[s,M]$ the maximal $p$-extension of $K$ with the Galois group of period $p^M$ and nilpotent class $s$. We apply the nilpotent Artin-Schreier theory together with the theory of the field-of-norms functor to give an explicit description of the Galois groups ${Gal}(K[s,M]/K)$. As application we prove that the ramification subgroup $Γ^{(v)}_K$ of the absolute Galois group of $K$ acts trivially on $K[s,M]$ if and only if $v>e_K(M+s/(p-1))-(1-δ_{1s})/p$, where $e_K$ is …
Using Kummer theory for a finite extension K of \Qp(ζ)(where p is a prime number and ζa primitive p-th root of~1), we compute the ramification filtration and the discriminant of an arbitrary elementary abelian p-extension of K. We also develop the analogous Artin-Schreier theory for finite extensions of \Fp((π)) and derive similar results for their elementary abelian p-extensions.
This paper contains three parts. In the first part, we give a thorough overview of the theory of Artin-Schreier-Witt extensions: this theory allows one to understand the $\mathbf{Z}/p^n\mathbf{Z}$-extensions of any field $K$ of characteristic $p$ via $p$-typical Witt vectors. Let $W_n(K)$ be the ring of $p$-typical Witt vectors of $K$ of length $n$ and let $\wp = F-\mathrm{id}: W_n(K)\longrightarrow W_n(K)$, where $F$ is the Frobenius map and $\mathrm{id}$ is the identity map. Artin-Schreier-Witt theory tells us that the abelian group $W_n(K)/\wp W_n(K)$ represents the set of $\mathbf{Z}/p^n\m…
If K_0 is the fraction field of the Witt vectors over an algebraically closed field k of characteristic p, we calculate upper bounds on the conductor of higher ramification for (the Galois closure of) extensions K_0(zeta_{p^r}, sqrt[p^r]{a})/K_0, where a is in K_0(zeta_{p^r}). Here zeta_{p^r} is a primitive p^r-th root of unity. In certain cases, including when a is in K_0 and p=2, we calculate the conductor exactly. These calculations can be used to determine the discriminants of various extensions of Q obtained by adjoining roots of unity and radicals.
One of the main results of this paper is a proof of the rank one case of an existence conjecture on lisse l-adic sheaves on a smooth variety over a finite field due to Deligne and Drinfeld. The problem is translated into the language of higher dimensional class field theory over finite fields, which describes the abelian fundamental group by Chow groups of zero cycles with moduli. A key ingredient is the construction of a cycle theoretic avatar of refined Artin conductor in ramification theory originally studied by Kazuya Kato.
We study the inequities in the distribution of Frobenius elements in Galois extensions of the rational numbers with Galois groups that are either dihedral $D_{2n}$ or (generalized) quaternion $\mathbb H_{2n}$ of two-power order. In the spirit of recent work of Fiorilli and Jouve arXiv:2001.05428, we study, under natural hypotheses, some families of such extensions, in a horizontal aspect, where the degree is fixed, and in a vertical aspect, where the degree goes to infinity. Our main contribution uncovers in families of extensions a phenomenon, for which Ng gave numerical evidence : real zeros…
A cohomological translation of the Kaplansky radical for profinite groups
Università di Milano-Bicocca（米兰比可卡大学）；Femto-St, Université Marie et Louis Pasteur（微纳科学与技术研究所，玛丽和路易·巴斯德大学）；Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）；University of Insubria（因斯布鲁克大学）
