Fields with two incomparable henselian valuation rings

Let K be a field and C, C' be two incomparable valuation rings of the separable closure of K, Theorem 1.2 states that the intersection of the decomposition groups of C, C', with respect to K, is precisely the inertia group of the composition ring C·C'. We apply this theorem in the study of two special cases of valued fields (L,B). In the first case, B is henselian and there is a subfield K of L such that L|K is a normal extension and B K is not henselian. The second case is that in which B has exactly two prolongations in the separable closure of L. We call these rings semihenselian rings, and they are characterized through Theorems 2.6 and 2.12.This paper is part of author's doctoral dissertation. Financial support for this research was provided by CNP_q (National Research Council) and by Universidade Estadual de Campinas.     

Approximating discrete valuation rings by regular local rings

Let be a field of characteristic zero and let be a discrete rank-one valuation domain containing with . Assume that the fraction field of has finite transcendence degree over . For every positive integer , we prove that can be realized as a directed union of regular local -subalgebras of of dimension .

     

Enumerating traceless matrices over compact discrete valuation rings

We show that the non-separable Banach space SL∞ is primary. This is achieved by directly solving the infinite-dimensional factorization problem in SL∞. In particular, we bypass Bourgain’s localization method.     

Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finite residue field $\mathsf{k}$ of odd characteristic, and let G be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $?\in \mathbb{N}$ let $G^{?}$ denote the ?-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{?+1}$ for some ?, and if its restriction to $G^{?}/G^{?+1}?\mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}({\mathsf{k}}^{\mathrm{alg}})$-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.     

Closed stability index of excellent henselian local rings

We show that the closed stability index of an excellent henselian local ring of real dimension d2 with real closed residue field is When d=2 it is shown that the value of can be either 2 or 3 and give characterizations of each of these values in terms of the relation of A with its normalization and in terms of the real spectrum of A.Mathematics Subject Classification (2000): 14P15, 32B10, 13J15, 13J25Partially supported by DGES BFM2002-04797 and EC contract HPRN-CT-2001-00271in final form: 15 October 2003     

On cuspidal representations of general linear groups over discrete valuation rings

We define a new notion of cuspidality for representations of GL _n over a finite quotient o _k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G _λ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of GL _n (F). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of GL _n (o _k ) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups G _λ . A functional equation for zeta functions for representations of GL _n (o _k ) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL₄(o₂) are constructed. Not all these representations are strongly cuspidal.     

Invariants of matrix pairs over discrete valuation rings and Littlewood-Richardson fillings

Let M and N be two r×r matrices of full rank over a discrete valuation ring R with residue field of characteristic zero. Let P,Q and T be invertible r×r matrices over R. It is shown that the orbit of the pair (M,N) under the action (M,N)?(PMQ^-1,QNT^-1) possesses a discrete invariant in the form of Littlewood-Richardson fillings of the skew shape λ/μ with content ν, where μ is the partition of orders of invariant factors of M, ν is the partition associated to N, and λ the partition of the product MN. That is, we may interpret Littlewood-Richardson fillings as a natural invariant of matrix pairs. This result generalizes invariant factors of a single matrix under equivalence, and is a converse of the construction in Appleby (1999) [1], where Littlewood-Richardson fillings were used to construct matrices with prescribed invariants. We also construct an example, however, of two matrix pairs that are not equivalent but still have the same Littlewood-Richardson filling. The filling associated to an orbit is determined by special quotients of determinants of a matrix in the orbit of the pair.     

Tolerant valuation rings

The recent book “Valued Fields” by A. J. Engler and A. Prestel contains the first monographic exposition of new important results (mostly due to J. Königsmann) on Henselian valued fields. By introducing a new notion of tolerant valuation rings, we offer a new look at the proofs of these results.     

Trees and valuation rings

A subring of a division algebra is called a valuation ring of if or holds for all nonzero in . The set of all valuation rings of is a partially ordered set with respect to inclusion, having as its maximal element. As a graph is a rooted tree (called the valuation tree of ), and in contrast to the commutative case, may have finitely many but more than one vertices. This paper is mainly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra , and one main result here is a positive answer to this question where can be chosen as a quaternion division algebra over a commutative field.