Tame characters and ramification of finite flat group schemes

In this paper, for a complete discrete valuation field K of mixed characteristic (0,p) and a finite flat group scheme G of p-power order over OK, we determine the tame characters appearing in the Galois representation in terms of the ramification theory of Abbes and Saito, without any restriction on the absolute ramification index of K or the embedding dimension of G.     

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 for j>u(K,r,n) acts trivially on the pn-torsion semi-stable GK-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.     

Ramification theory for varieties over a local field

We define generalizations of classical invariants of wild ramification for coverings on a variety of arbitrary dimension over a local field. For an ?-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 character for a group action with an isolated fixed point on a regular local ring, assuming the dimension is 2.     

On saturated distinguished chains over a local field II

This is a continuation of the paper (J Number Theory 79 (1999) 217) on saturated distinguished chains (SDCs) over a local field K. Here, we give a “canonical” choice of the next element α₁ in a SDC for α=π₁+π₁π considered in (Ota, 1999) for wildly totally ramified Galois extensions, and this leads us to consider a tower of fields, K⊂K₁⊂K₂⊂…, where K₁=K(α₁) and K_n/K is wildly totally ramified. The union of these fields is particularly interesting, for its conductor over K is very small, close to 1. Moreover, in some cases K is uniquely determined up to isomorphism over K for any such extensions of the same type. We also consider SDCs for an element α=π₁+π₁π₂+?+π₁π₂?π_n for totally ramified Galois extensions of type (m,m,...,m), where m is a power of the characteristic of the residual field of K.     

Ramification of valuations

In this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields. In characteristic zero and arbitrary dimension, we obtain the strongest possible generalization of the classical ramification theory of local Dedekind domains. We further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic. We prove that local monomialization and simultaneous resolution hold under very mild assumptions, and give pathological examples.     

Rational points and Galois points for a plane curve over a finite field

We study the relationship between rational points and Galois points for a plane curve over a finite field. It is known that the set of Galois points coincides with that of rational points of the projective plane if the curve is the Hermitian, Klein quartic or Ballico–Hefez curve. The author proposes a problem: Does the converse hold true? If the curve of genus zero or one has a rational point, we have an affirmative answer.     

Computing irreducible representations of supersolvable groups over small finite fields

We present an algorithm to compute a full set of irreducible representations of a supersolvable group over a finite field , , which is not assumed to be a splitting field of . The main subroutines of our algorithm are a modification of the algorithm of Baum and Clausen (Math. Comp. 63 (1994), 351-359) to obtain information on algebraically conjugate representations, and an effective version of Speiser's generalization of Hilbert's Theorem 90 stating that vanishes for all .

     

Ramification groups of infinite p-extensions of a local field

The upper enumeration of ramification groups for certain infinite p -extensions is studied. In particular, the sequence of ramification groups is completely calculated for a simple class of non-Abelian extensions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 96–132, 1977.     

On the genus of a cyclic curve over a finite field

Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer type is computed.     

Ramification of Valuations and Local Rings in Positive Characteristic

In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension.     

Symmetric bilinear forms over finite fields of even characteristic

Let Sm be the set of symmetric bilinear forms on an m-dimensional vector space over GF(q), where q is a power of two. A subset Y of Sm is called an (m,d)-set if the difference of every two distinct elements in Y has rank at least d. Such objects are closely related to certain families of codes over Galois rings of characteristic four. An upper bound on the size of (m,d)-sets is derived, and in certain cases, the rank distance distribution of an (m,d)-set is explicitly given. Constructions of (m,d)-sets are provided for all possible values of m and d.     

Subregular characters of the unitriangular group over a finite field

We prove a formula for subregular characters of the unitriangular group over a finite field in terms of coefficients of minors of the characteristic matrix. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 5, pp. 103–125, 2007.     

Counting irreducible polynomials with prescribed coefficients over a finite field

We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by equivalence classes of polynomials with prescribed coefficients. Simplified expressions are derived for some special cases. Our results extend some earlier results.     

On the essential dimension of a finite group

Let f(x) = aixi be a monic polynomial of degree n whosecoefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) isgenerating (for the symmetric group) if it can be obtained from f(x) by a nondegenerate Tschirnhaus transformation. We show that the minimal number dk(n) of algebraically independent coefficients of such a polynomial is at least [n/2]. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilberts 13th problem.Our approach to this question (and generalizations) is basedon the idea of the essential dimension of a finite group G:the smallest possible dimension of an algebraic G-variety over k to which one can compress a faithful linear representation of G. We show that dk(n) is just the essential dimension of the symmetricgroup Sn. We give results on the essential dimension ofother groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.