scholarly journals Rankin–Selberg gamma factors of level zero representations of GLn

2019 ◽  
Vol 31 (2) ◽  
pp. 503-516 ◽  
Author(s):  
Rongqing Ye
Keyword(s):  
Local Field ◽  
Residue Field ◽  
Converse Theorem ◽  
Level Zero ◽  
Gamma Factor ◽  
Cuspidal Representations

AbstractFor a p-adic local field F of characteristic 0, with residue field {\mathfrak{f}}, we prove that the Rankin–Selberg gamma factor of a pair of level zero representations of linear general groups over F is equal to a gamma factor of a pair of corresponding cuspidal representations of linear general groups over {\mathfrak{f}}. Our results can be used to prove a variant of Jacquet’s conjecture on the local converse theorem.

scholarly journals On classification of typical representations for GL3(F)

2019 ◽  
Vol 31 (4) ◽  
pp. 917-941
Author(s):  
Santosh Nadimpalli
Keyword(s):  
Local Field ◽  
Residue Field ◽  
Principal Series

Abstract Let F be any non-Archimedean local field with residue field of cardinality {q_{F}} . In this article, we obtain a classification of typical representations for the Bernstein components associated to the inertial classes of the form {[\operatorname{GL}_{n}(F)\times F^{\times},\sigma\otimes\chi]} with {q_{F}>2} , and for the principal series components with {q_{F}>3} . With this we complete the classification of typical representations for {\operatorname{GL}_{3}(F)} , for {q_{F}>2} .

scholarly journals Local–global principle for reduced norms over function fields of -adic curves

2017 ◽  
Vol 154 (2) ◽  
pp. 410-458 ◽  
Author(s):  
R. Parimala ◽  
R. Preeti ◽  
V. Suresh
Keyword(s):  
Local Field ◽  
Function Field ◽  
Function Fields ◽  
Residue Field ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Hasse Principle ◽  
Central Simple ◽  
Global Principle ◽  
Reduced Norm

Let $K$ be a (non-archimedean) local field and let $F$ be the function field of a curve over $K$. Let $D$ be a central simple algebra over $F$ of period $n$ and $\unicode[STIX]{x1D706}\in F^{\ast }$. We show that if $n$ is coprime to the characteristic of the residue field of $K$ and $D\cdot (\unicode[STIX]{x1D706})=0$ in $H^{3}(F,\unicode[STIX]{x1D707}_{n}^{\otimes 2})$, then $\unicode[STIX]{x1D706}$ is a reduced norm from $D$. This leads to a Hasse principle for the group $\operatorname{SL}_{1}(D)$, namely, an element $\unicode[STIX]{x1D706}\in F^{\ast }$ is a reduced norm from $D$ if and only if it is a reduced norm locally at all discrete valuations of $F$.

scholarly journals Wild ramification and the characteristic cycle of an ℓ-adic sheaf

2009 ◽  
Vol 8 (4) ◽  
pp. 769-829 ◽  
Author(s):  
Takeshi Saito
Keyword(s):  
Local Field ◽  
Cotangent Bundle ◽  
Euler Number ◽  
Residue Field ◽  
Geometric Method ◽  
Characteristic Class ◽  
Wild Ramification ◽  
Characteristic Cycle

AbstractWe propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an ℓ-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent bundle and prove that the intersection with the 0-section computes the characteristic class, and hence the Euler number.

scholarly journals Stability of Asai local factors for GL(2)

2020 ◽  
pp. 1-18
Author(s):  
Yeongseong Jo ◽  
M. Krishnamurthy
Keyword(s):  
Local Field ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Quadratic Algebra ◽  
Local Factors ◽  
The Stability ◽  
Gamma Factor

Let [Formula: see text] be a non-archimedean local field of characteristic not equal to 2 and let [Formula: see text] be a quadratic algebra. We prove the stability of local factors attached to irreducible admissible (complex) representations of [Formula: see text] via the Rankin–Selberg method under highly ramified twists. This includes both the Asai as well as the Rankin–Selberg local factors attached to pairs. Our method relies on expressing the gamma factor as a Mellin transform using Bessel functions.

scholarly journals On the action of the unitary group on the projective plane over a local field

1997 ◽  
Vol 62 (3) ◽  
pp. 371-397 ◽  
Author(s):  
Harm Voskuil
Keyword(s):  
Projective Plane ◽  
Local Field ◽  
Unitary Group ◽  
Residue Field ◽  
Equivariant Map ◽  
Rank One ◽  
Characteristic 2 ◽  
Set Of Points ◽  
Stable Points ◽  
Split Torus

AbstractLet G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ≠ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.

Generic cuspidal representations of 𝑈(2, 1)

2021 ◽  
Vol 33 (3) ◽  
pp. 709-742
Author(s):  
Santosh Nadimpalli
Keyword(s):  
Local Field ◽  
Classical Groups ◽  
Fixed Field ◽  
Residue Characteristic ◽  
Cuspidal Representations

Abstract Let 𝐹 be a non-Archimedean local field, and let 𝜎 be a non-trivial Galois involution with fixed field F 0 F_{0} . When the residue characteristic of F 0 F_{0} is odd, using the construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of U ⁢ ( 2 , 1 ) ⁢ ( F / F 0 ) U(2,1)(F/F_{0}) .

scholarly journals Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of 𝔭-Adic Simple Algebras

2003 ◽  
Vol 55 (2) ◽  
pp. 353-378 ◽  
Author(s):  
Allan J. Silberger ◽  
Ernst-Wilhelm Zink
Keyword(s):  
Local Field ◽  
Discrete Series ◽  
Unit Group ◽  
Level Zero ◽  
Series Representations ◽  
Matching Theorem ◽  
Discrete Series Representations ◽  
Simple Algebras ◽  
Central Simple

AbstractLet F be a p-adic local field and let be the unit group of a central simple F-algebra Ai of reduced degree n > 1 (i = 1, 2). Let ℛ2() denote the set of irreducible discrete series representations of . The “Abstract Matching Theorem” asserts the existence of a bijection, the “Jacquet- Langlands” map, which, up to known sign, preserves character values for regular elliptic elements. This paper addresses the question of explicitly describing the map 𝒥ℒ, but only for “level zero” representations. We prove that the restriction is a bijection of level zero discrete series (Proposition 3.2) and we give a parameterization of the set of unramified twist classes of level zero discrete series which does not depend upon the algebra Ai and is invariant under 𝒥ℒA2,A1 (Theorem 4.1).

scholarly journals A family of Eisenstein polynomials generating totally ramified extensions, identification of extensions and construction of class fields

2014 ◽  
Vol 10 (07) ◽  
pp. 1699-1727 ◽  
Author(s):  
Maurizio Monge
Keyword(s):  
Normal Form ◽  
Local Field ◽  
Isomorphism Class ◽  
Residue Field ◽  
Class Field ◽  
Isomorphism Classes ◽  
Special Polynomials ◽  
Special Equation ◽  
The Family ◽  
Class Fields

Let K be a local field with finite residue field, we define a normal form for Eisenstein polynomials depending on the choice of a uniformizer πK and of residue representatives. The isomorphism classes of extensions generated by the polynomials in the family exhaust all totally ramified extensions, and the multiplicity with which each isomorphism class L/K appears is always smaller than the number of conjugates of L over K. An algorithm to recover the set of all special polynomials generating the extension determined by a general Eisenstein polynomial is described. We also give a criterion to quickly establish that a polynomial generates a different extension from that generated by a set of special polynomials, such criterion does not only depend on the usual distance on the set of Eisenstein polynomials considered by Krasner and others. We conclude with an algorithm for the construction of the unique special equation determining a totally ramified class field in general degree, given a suitable representation of a group of norms.

Solvable Points on Projective Algebraic Curves

2004 ◽  
Vol 56 (3) ◽  
pp. 612-637 ◽  
Author(s):  
Ambrus Pál
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Algebraic Curves ◽  
Residue Field ◽  
Rational Points ◽  
Projective Curve ◽  
Arbitrary Characteristic ◽  
Projective Curves ◽  
Solvable Extension ◽  
Solvable Points

AbstractWe examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus g, when g is at least 40, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus 0, 2, 3 or 4 defined over any field has a solvable point. Finally we prove that every genus 1 curve defined over a local field of characteristic zero with residue field of characteristic p has a divisor of degree prime to 6p defined over a solvable extension.

Some Orbital Integrals and a Technique for Counting Representations of GL 2(F)

1978 ◽  
Vol 30 (02) ◽  
pp. 431-448 ◽  
Author(s):  
T. Callahan
Keyword(s):  
Haar Measure ◽  
Local Field ◽  
Maximal Ideal ◽  
Characteristic Zero ◽  
Maximal Compact Subgroup ◽  
Residue Field ◽  
Compact Subgroup ◽  
Orbital Integrals ◽  
Ring Of Integers

Let F be a local field of characteristic zero, with q elements in its residue field, ring of integers uniformizer ωF and maximal ideal . Let GF = GL2(F). We fix Haar measures dg and dz on GF and ZF, the centre of GF, so that meas(K) = meas where K = GL2() is a maximal compact subgroup of GF. If T is a torus in GF we take dt to be the Haar measure on T such that means(TM)=1 where TM denotes the maximal compact subgroup of T.

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