scholarly journals Le lemme fondamental pondéré. I. Constructions géométriques

2010 ◽  
Vol 146 (6) ◽  
pp. 1416-1506 ◽  
Author(s):  
Pierre-Henri Chaudouard ◽  
Gérard Laumon
Keyword(s):  
Finite Field ◽  
Affine Space ◽  
Base Space ◽  
Characteristic Polynomials ◽  
Selberg Trace Formula ◽  
Orbital Integrals ◽  
Fundamental Lemma ◽  
Open Set ◽  
Simple Set ◽  
Hitchin Fibration

AbstractThis work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu’s proof of the Langlands–Shelstad fundamental lemma. Ngô’s approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of ‘characteristic polynomials’. Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fibration over an open set larger than the elliptic set, namely the ‘generically regular semi-simple set’. The fibers are in general neither of finite type nor separated. By analogy with Arthur’s truncation, we introduce the substack of ξ-stable Hitchin bundles. We show that it is a Deligne–Mumford stack, smooth over the base field and proper over the base space of ‘characteristic polynomials’. Moreover, the number of points of the ξ-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur–Selberg trace formula.

scholarly journals Transfert d’intégrales orbitales pour le groupe métaplectique

2010 ◽  
Vol 147 (2) ◽  
pp. 524-590 ◽  
Author(s):  
Wen-Wei Li
Keyword(s):  
Trace Formula ◽  
Local Field ◽  
Transfer Factor ◽  
Characteristic Zero ◽  
Prior Work ◽  
Selberg Trace Formula ◽  
Metaplectic Groups ◽  
Orbital Integrals ◽  
Fundamental Lemma ◽  
Set Up

AbstractWe set up a formalism of endoscopy for metaplectic groups. By defining a suitable transfer factor, we prove an analogue of the Langlands–Shelstad transfer conjecture for orbital integrals over any local field of characteristic zero, as well as the fundamental lemma for units of the Hecke algebra in the unramified case. This generalizes prior work of Adams and Renard in the real case and serves as a first step in studying the Arthur–Selberg trace formula for metaplectic groups.

scholarly journals On the class of square Petrie matrices induced by cyclic permutations

2004 ◽  
Vol 2004 (31) ◽  
pp. 1617-1622
Author(s):  
Bau-Sen Du
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Topological Entropy ◽  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Dynamical Properties

Letn≥2be an integer and letP={1,2,…,n,n+1}. LetZpdenote the finite field{0,1,2,…,p−1}, wherep≥2is a prime. Then every mapσonPdetermines a realn×nPetrie matrixAσwhich is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization ofσ. In this paper, we show that ifσis acyclicpermutation onP, then all such matricesAσare similar to one another overZ2(but not overZpfor any primep≥3) and their characteristic polynomials overZ2are all equal to∑k=0nxk. As a consequence, we obtain that ifσis acyclicpermutation onP, then the coefficients of the characteristic polynomial ofAσare all odd integers and hence nonzero.

On the Fundamental Lemma for Standard Endoscopy: Reduction to Unit Elements

1995 ◽  
Vol 47 (5) ◽  
pp. 974-994 ◽  
Author(s):  
Thomas C. Hales
Keyword(s):  
Trace Formula ◽  
Simple Form ◽  
Hecke Algebras ◽  
Orbital Integrals ◽  
Fundamental Lemma ◽  
Stable Trace Formula

AbstractThe fundamental lemma for standard endoscopy follows from the matching of unit elements in Hecke algebras. A simple form of the stable trace formula, based on the matching of unit elements, shows the fundamental lemma to be equivalent to a collection of character identities. These character identities are established by comparing them to a compact-character expansion of orbital integrals.

Linearized systems and a generalized Cayley–Hamilton theorem

2017 ◽  
Vol 16 (06) ◽  
pp. 1750120
Author(s):  
Jeffrey Lang ◽  
Daniel Newland
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Minimum Distance ◽  
Solution Space ◽  
Arbitrary Field ◽  
Characteristic Polynomials ◽  
Systems Of Equations ◽  
Square Matrix

We study linearized systems of equations in characteristic [Formula: see text] of the form [Formula: see text] where [Formula: see text] is a square matrix and [Formula: see text]. We present algorithms for calculating their solutions and for determining the minimum distance of their solution spaces. In the case when [Formula: see text] has entries in [Formula: see text], the finite field of [Formula: see text] elements, we explore the relationships between the minimal and characteristic polynomials of [Formula: see text] and the above mentioned features of the solution space. In order to extend and generalize these findings to the case when [Formula: see text] has entries in an arbitrary field of characteristic [Formula: see text], we obtain generalizations of the characteristic polynomial of a matrix and the Cayley–Hamilton theorem to square linearized systems.

scholarly journals Computing the Characteristic Polynomials of a Class of Hyperelliptic Curves for Cryptographic Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Lin You ◽  
Guangguo Han ◽  
Jiwen Zeng ◽  
Yongxuan Sang
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Hyperelliptic Curve ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Common Method ◽  
Characteristic Polynomials ◽  
Computational Data ◽  
Prime Factors

Hyperelliptic curves have been widely studied for cryptographic applications, and some special hyperelliptic curves are often considered to be used in practical cryptosystems. Computing Jacobian group orders is an important operation in constructing hyperelliptic curve cryptosystems, and the most common method used for the computation of Jacobian group orders is by computing the zeta functions or the characteristic polynomials of the related hyperelliptic curves. For the hyperelliptic curveCq:v2=up+au+bover the fieldFqwithqbeing a power of an odd primep, Duursma and Sakurai obtained its characteristic polynomial forq=p,a=−1,andb∈Fp. In this paper, we determine the characteristic polynomials ofCqover the finite fieldFpnforn=1, 2 anda,b∈Fpn. We also give some computational data which show that many of those curves have large prime factors in their Jacobian group orders, which are both practical and vital for the constructions of efficient and secure hyperelliptic curve cryptosystems.

scholarly journals CHARACTERISTIC POLYNOMIALS OF SIMPLE ORDINARY ABELIAN VARIETIES OVER FINITE FIELDS

2021 ◽  
pp. 1-7
Author(s):  
LENNY JONES
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Abelian Varieties ◽  
Characteristic Polynomials ◽  
Easy Method

Abstract We provide an easy method for the construction of characteristic polynomials of simple ordinary abelian varieties ${{\mathcal A}}$ of dimension g over a finite field ${{\mathbb F}}_q$ , when $q\ge 4$ and $2g=\rho ^{b-1}(\rho -1)$ , for some prime $\rho \ge 5$ with $b\ge 1$ . Moreover, we show that ${{\mathcal A}}$ is absolutely simple if $b=1$ and g is prime, but ${{\mathcal A}}$ is not absolutely simple for any prime $\rho \ge 5$ with $b>1$ .

scholarly journals Assignments to sheaves of pseudometric spaces

2020 ◽  
Vol 2 ◽  
pp. 2
Author(s):  
Michael Robinson
Keyword(s):  
Noisy Data ◽  
Base Space ◽  
Local Section ◽  
Structure Preserving ◽  
Continuous Map ◽  
Open Set

An assignment to a sheaf is the choice of a local section from each open set in the sheaf's base space, without regard to how these local sections are related to one another. This article explains that the consistency radius --- which quantifies the agreement between overlapping local sections in the assignment --- is a continuous map. When thresholded, the consistency radius produces the consistency filtration, which is a filtration of open covers. This article shows that the consistency filtration is a functor that transforms the structure of the sheaf and assignment into a nested set of covers in a structure-preserving way. Furthermore, this article shows that consistency filtration is robust to perturbations, establishing its validity for arbitrarily thresholded, noisy data.

On Conics Over a Finite Field

1974 ◽  
Vol 26 (6) ◽  
pp. 1281-1288
Author(s):  
Fuanglada R. Jung
Keyword(s):  
Finite Field ◽  
Affine Space ◽  
Galois Field ◽  
Base Field ◽  
Image Position

Let F denote a Galois field of order q and odd characteristic p, and F* = F\{0}. Let Sn denote an n-dimensional affine space with base field F. E. Cohen [1] had proved that if n ≧ 4, there is no hyperplane of Sn contained in the complement of the quadric Qn of Sn defined by1.1and in S3, there are q + 1 or 0 planes contained in the complement of Q3 according as — aα is not or is a square of F.

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