Ramified Extensions Research Articles

On the compositum of wildly ramified extensions

The ramification filtration on the compositum of two wildly ramified extensions is computed in various cases. Some positive results towards Abhyankar's inertia conjecture have been also proved.

Ramification correspondence of finite flat group schemes over equal and mixed characteristic local fields

Let p>2 be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fractional field of the Witt ring of k. Let G and H be finite flat commutative group schemes killed by p over O_K and k[[u]], respectively. In this paper, we show the upper and the lower ramification subgroups of G and H in the sense of Abbes-Saito are naturally isomorphic to each other when they are associated to the same Kisin module.

Models of group schemes of roots of unity

Let O_K be a discrete valuation ring of mixed characteristics (0,p), with residue field k. Using work of Sekiguchi and Suwa, we construct some finite flat O_K-models of the group scheme \mu_{p^n,K} of p^n-th roots of unity, which we call Kummer group schemes. We set carefully the general framework and algebraic properties of this construction. When k is perfect and O_K is a complete totally ramified extension of the ring of Witt vectors W(k), we provide a parallel study of the Breuil-Kisin modules of finite flat models of \mu_{p^n,K}, in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. We compute these objects for n < 4. This leads us to conjecture that all finite flat models of \mu_{p^n,K} are Kummer group schemes.

Leading coefficients of isogenies of degree $p$ over $\mathbb{Q}_{p}$

Let E be an elliptic curve over p which has potentially supersingular good reduction. Let L p be a totally ramified extension such that E has good reduction over L and E be the reduction of E mod , where is a prime element of the ring of integers OL of L. Let E be the formal group over OL associated to E OL . The multiplication by p map [p] E E is written by power series [ p](x) px c2x 2 c px p c p2 x p 2 O L[[x]]. By using the liftings over OL of the Dieudonne module of p-divisible group E( p) over p, we determine the values of L(c p).

NON-INVARIANCE OF THE INDEX IN WILDLY RAMIFIED EXTENSIONS

In this paper, we give an example of three wildly ramified extensions L1, L2, L3 of ℚ2 with the same ramification numbers and isomorphic Galois groups, such that I(nL1) &gt; I(nL2) &gt; I(nL3) for a suitable integer n (where I(nL) denotes the index of the ℚ2-algebra Ln). This example shows that the condition given in [2] for the invariance of the index of tamely ramified extensions is no longer sufficient in the general case.

Moduli of finite flat group schemes, and modularity

We prove that, under some mild conditions, a two dimensional p-adic Galois representation which is residually modular and potentially Barsotti-Tate at p is modular. This provides a more conceptual way of establishing the Shimura-Taniyama-Weil conjecture, especially for elliptic curves which acquire good reduction over a wildly ramified extension of ℚ 3 . The main ingredient is a new technique for analyzing flat deformation rings. It involves resolving them by spaces which parametrize finite flat group scheme models of Galois representations. Introduction.

On the construction of normal wildly ramified extensions over Q p , (p ≠ 2)

On the construction of normal wildly ramified extensions over Q p , (p ≠ 2)

On the construction of normal wildly ramified extensions over Q 2

It is well known that a finite totally ramified extension of a local field can be generated by a uniformising element the minimal polynomial of which is also Eisenstein. The quadratic and the quartic normal totally ramified extensions of Q2 are well known and well characterized. In this note we characterize the Eisenstein polynomials of degree 4 with coefficients in Z2 that define normal totally ramified extensions of Q2. Furthermore we give some necessary conditions for the cyclic case of degree 2n. Also examples are given.

On a ramification bound of torsion semi-stable representations over a local field

Let p be a rational prime, k be a perfect field of characteristic p, W = W ( k ) be the ring of Witt vectors, K be a finite totally ramified extension of Frac ( W ) of degree e and r be a non-negative integer satisfying r < p − 1 . In this paper, we prove the upper numbering ramification group G K ( j ) for j > u ( K , r , n ) acts trivially on the p n -torsion semi-stable G K -representations with Hodge–Tate weights in { 0 , … , r } , where u ( K , 0 , n ) = 0 , u ( K , 1 , n ) = 1 + e ( n + 1 / ( p − 1 ) ) and u ( K , r , n ) = 1 − p − n + e ( n + r / ( p − 1 ) ) for 1 < r < p − 1 .

Non-Abelian associated orders of wildly ramified extensions

In the article, a mixed characteristic wildly ramified local field extension is considered. The structure of the associated order of such an extension and the question of whether an extension with a given associated order exists are studied. The non-Abelian case is described. A classification theorem is proved. A series of non-Abelian extensions of a given type is constructed. Bibliography: 12 titles.

On the Structure of Some Moduli Spaces of Finite Flat Group Schemes

Let p be an odd prime and k a finite field of characteristic p. Let W = W (k) be the ring of Witt vectors with coefficients in k and K0 = W [ 1 p ] its fraction field. We consider a finite, totally ramified extension K/K0 and denote by e = [K : K0] the degree of the extension. Let us fix a uniformizer π ∈ OK with minimal polynomial E(u) ∈ W [u] over K0. Further we fix an algebraic closure K of K. Let F be a finite field of characteristic p and ρ : GK → GL(VF) a continuous representation of the absolute Galois group GK = Gal(K/K) of K in a finite dimensional F-vector space VF whose dimension will be denoted by d. This datum is equivalent to a finite commutative group scheme G → Spec K with

Breuil modules for Raynaud schemes

Let p be an odd prime, and let O K be the ring of integers in a finite extension K / Q p . Breuil has classified finite flat group schemes of type ( p , … , p ) over O K in terms of linear-algebraic objects that have come to be known as Breuil modules. This classification can be extended to the case of finite flat vector space schemes G over O K . When G has rank one, the generic fiber of G corresponds to a Galois character, and we explicitly determine this character in terms of the Breuil module of G . Special attention is paid to Breuil modules with descent data corresponding to characters of Gal ( Q ¯ p / Q p d ) that become finite flat over a totally ramified extension of degree p d − 1 ; these arise in Gee's work on the weight in Serre's conjecture over totally real fields. Video abstract For a video summary of this paper, please visit http://www.youtube.com/watch?v=9oWYJy_XrZE.

Leopoldt's problem for Abelian totally ramified extensions of complete discrete valuation fields

By using methods described in earlier papers of the author, it is proved that, in many cases, if an Abelian totally ramified p-extension contains an ideal free over its associated order, then the extension is of the type described and completely classified in an earlier paper of the author (such extensions are said to be semistable). A counterexample to this statement is presented in the case where the conditions on the extension are not fulfilled. Several other properties of extensions in question are proved.

Motivic Serre invariants of curves

In this article, we generalize the theory of motivic integration on formal schemes topologically of finite type and the notion of motivic Serre invariant, to a relative point of view. We compute the relative motivic Serre invariant for curves defined over the field of fractions of a complete discrete valuation ring R of equicharacteristic zero. One aim of this study is to understand the behavior of motivic Serre invariants under ramified extension of the ring R. Thanks to our constructions, we obtain, in particular, an expression for the generating power series, whose coefficients are the motivic Serre invariant associated to a curve, computed on a tower of ramified extensions of R. We give an interpretation of this series in terms of the motivic zeta function of Denef and Loeser.

The Cech filtration and monodromy in log crystalline cohomology

For a strictly semistable log scheme Y Y over a perfect field k k of characteristic p p we investigate the canonical Čech spectral sequence ( C ) T (C)_T abutting the Hyodo-Kato (log crystalline) cohomology H c r y s ∗ ( Y / T ) Q H_{crys}^*(Y/T)_{\mathbb {Q}} of Y Y and beginning with the log convergent cohomology of its various component intersections Y i Y^i . We compare the filtration on H c r y s ∗ ( Y / T ) Q H_{crys}^*(Y/T)_{\mathbb {Q}} arising from ( C ) T (C)_T with the monodromy operator N N on H c r y s ∗ ( Y / T ) Q H_{crys}^*(Y/T)_{\mathbb {Q}} . We also express N N through residue maps and study relations with singular cohomology. If Y Y lifts to a proper strictly semistable (formal) scheme X X over a finite totally ramified extension of W ( k ) W(k) , with generic fibre X K X_K , we obtain results on how the simplicial structure of X K a n X_K^{an} (as analytic space) is reflected in H d R ∗ ( X K ) = H d R ∗ ( X K a n ) H_{dR}^*(X_K)=H_{dR}^*(X_K^{an}) .

Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants

Using Serre's mass formula [15] for totally ramified extensions, we derive a mass formula that counts all (isomorphism classes of) étale algebra extensions of a local field F having a given degree n. Along the way, we also prove a series of mass formulae for counting étale extensions of a local field F having certain properties, such as a chosen prime splitting or ramification behavior. We then use these mass formulae to formulate a heuristic that predicts the asymptotic number of number fields, of fixed degree n and bounded discriminant, whose Galois closures have Galois group the full symmetric group Sn over ℚ. Analogous predictions are made for the asymptotic density of those number fields having specified local behaviors at finitely many places. All these predictions are in full agreement with the known results in degrees up to five.

The Compositum of Wild Extensions of Local Fields of Prime Degree

In this paper we present a general view of the totally and wildly ramified extensions of degree p of a p-adic field K. Our method consists in deducing the properties of the set of all extensions of degree p of K from the study of the compositum ${\cal C}_K(p)$ of all its elements. We show that in fact ${\cal C}_K(p)$ is the maximal abelian extension of exponent p of F = F(K), where F is the compositum of all cyclic extensions of K of degree dividing p − 1. By our method, it is fairly simple to recover the distribution of the extensions of K of degree p (and also of their isomorphism classes) according to their discriminant.

ON SATURATED DISTINGUISHED CHAINS OVER A LOCAL FIELD III: The Case of p-Extensions with Two Proper Higher Ramification Groups

This is a continuation of the papers [5, 6] on saturated distinguished chains (SDCs) over a local field K. We consider a totally ramified Galois extension L of K whose Galois group is a p-group and has exactly two proper higher ramification groups, where p is the characteristic of the residual field. Take a prime element π of L, and set πi = NL/Mi(π), where Mi are the corresponding fields to higher ramification groups such that K ⊂ M1 ⊂ M2 ⊂ L and NL/Mi denotes the norm map from L to Mi for each i. We consider a SDC for α = π1 + π1π2 + π1π2π over K, and the result is given as Main Theorem. There are thirteen cases according to sizes of extension degrees [L : M2], [M2 : M1] and [M1 : K].

Dynamical systems in unramified or totally ramified extensions of a -adic field

We use local field theory to study a special class of discrete dynamical systems, where the function being iterated is a polynomial whose coefficients belong to the ring of integers in a -adic field.

Nicely semiramified division algebras over Henselian fields

This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite‐dimensional central division algebra over a Henselian field E with an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions of E) is nicely semiramified.