scholarly journals Compatible systems and ramification

2019 ◽  
Vol 155 (12) ◽  
pp. 2334-2353 ◽  
Author(s):  
Qing Lu ◽  
Weizhe Zheng
Keyword(s):  
Finite Field ◽  
Finite Type ◽  
Local Field ◽  
Ring Of Integers

We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on $\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

On the Integral Extensions of Quadratic Forms Over Local Fields

1970 ◽  
Vol 22 (2) ◽  
pp. 297-307 ◽  
Author(s):  
Melvin Band
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Ring Of Integers ◽  
Finite Dimensional ◽  
Integral Analogue ◽  
Necessary And Sufficient ◽  
Special Case

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

Ramification Groups of Abelian Local Field Extensions

1971 ◽  
Vol 23 (2) ◽  
pp. 271-281 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Galois Extension ◽  
Residue Class ◽  
Abelian Extensions ◽  
Ring Of Integers ◽  
Field Extensions ◽  
Valued Field ◽  
Residue Class Field ◽  
Image Position

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

LOGICAL CHARACTERIZATION OF RECOGNIZABLE SETS OF POLYNOMIALS OVER A FINITE FIELD

2011 ◽  
Vol 22 (07) ◽  
pp. 1549-1563 ◽  
Author(s):  
MICHEL RIGO ◽  
LAURENT WAXWEILER
Keyword(s):  
Finite Field ◽  
Order Theory ◽  
Ring Of Integers ◽  
First Order ◽  
Polynomial Coefficients ◽  
Bounded Number ◽  
First Order Theory ◽  
Ring Of Polynomials

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination of powers of a non-constant polynomial P playing the role of the base of the numeration. Having in mind the theorem of Cobham from 1969 about recognizable sets of integers, it is natural to study P-recognizable sets of polynomials. Based on the results obtained in the Ph.D. thesis of the second author, we study the logical characterization of such sets and related properties like decidability of the corresponding first-order theory.

Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field

2019 ◽  
Vol 52 (1) ◽  
pp. 59-65
Author(s):  
A. I. Madunts ◽  
S. V. Vostokov ◽  
R. P. Vostokova
Keyword(s):  
Local Field ◽  
Formal Groups ◽  
Ring Of Integers

scholarly journals Multiplicity Free Jacquet Modules

2012 ◽  
Vol 55 (4) ◽  
pp. 673-688 ◽  
Author(s):  
Avraham Aizenbud ◽  
Dmitry Gourevitch
Keyword(s):  
Finite Field ◽  
Natural Number ◽  
Parabolic Subgroup ◽  
Local Field ◽  
Classical Method ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Unipotent Subgroup ◽  
Multiplicity Free

AbstractLet F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G := GLn+k(F) and let M := GLn(F) × GLk(F) < G be a maximal Levi subgroup. Let U < G be the corresponding unipotent subgroup and let P = MU be the corresponding parabolic subgroup. Let be the Jacquet functor, i.e., the functor of coinvariants with respect toU. In this paper we prove that J is a multiplicity free functor, i.e., dim HomM(J(π), ρ) ≤ 1, for any irreducible representations π of G and ρ of M. We adapt the classical method of Gelfand and Kazhdan, which proves the “multiplicity free” property of certain representations to prove the “multiplicity free” property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.

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