scholarly journals STABLE MODELS OF LUBIN–TATE CURVES WITH LEVEL THREE

2016 ◽  
Vol 225 ◽  
pp. 100-151 ◽  
Author(s):  
NAOKI IMAI ◽  
TAKAHIRO TSUSHIMA
Keyword(s):  
Division Algebra ◽  
Local Field ◽  
Formal Model ◽  
Residue Field ◽  
Weil Group ◽  
Stable Reduction ◽  
Tate Curve ◽  
Stable Models

We construct a stable formal model of a Lubin–Tate curve with level three, and study the action of a Weil group and a division algebra on its stable reduction. Further, we study a structure of cohomology of the Lubin–Tate curve. Our study is purely local and includes the case where the characteristic of the residue field of a local field is two.

SLn, Orthogonality Relations and Transfer

2007 ◽  
Vol 59 (3) ◽  
pp. 449-464 ◽  
Author(s):  
Alexandru Ioan Badulescu
Keyword(s):  
Division Algebra ◽  
Local Field ◽  
Finite Dimension ◽  
Complex Conjugate ◽  
Orbital Integrals ◽  
Central Division ◽  
Zero Characteristic ◽  
Central Division Algebra ◽  
Orthogonality Relations ◽  
Square Integrable Representation

AbstractLet π be a square integrable representation of G′ = SLn(D), with D a central division algebra of finite dimension over a local field F of non-zero characteristic. We prove that, on the elliptic set, the character of π equals the complex conjugate of the orbital integral of one of the pseudocoefficients of π. We prove also the orthogonality relations for characters of square integrable representations of G′. We prove the stable transfer of orbital integrals between SLn(F) and its inner forms.

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 THE -MODULAR LOCAL LANGLANDS CORRESPONDENCE AND LOCAL CONSTANTS

2019 ◽  
pp. 1-51 ◽  
Author(s):  
R. Kurinczuk ◽  
N. Matringe
Keyword(s):  
Prime Number ◽  
Local Field ◽  
Equivalence Classes ◽  
Modular Representations ◽  
Langlands Correspondence ◽  
Weil Group ◽  
Local Langlands Correspondence ◽  
Residual Characteristic

Let  $F$ be a non-archimedean local field of residual characteristic  $p$ , $\ell \neq p$ be a prime number, and  $\text{W}_{F}$ the Weil group of  $F$ . We classify equivalence classes of  $\text{W}_{F}$ -semisimple Deligne  $\ell$ -modular representations of  $\text{W}_{F}$ in terms of irreducible  $\ell$ -modular representations of  $\text{W}_{F}$ , and extend constructions of Artin–Deligne local constants to this setting. Finally, we define a variant of the  $\ell$ -modular local Langlands correspondence which satisfies a preservation of local constants statement for pairs of generic representations.

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.

On the Restriction to 𝒟* × 𝒟* of Representations of p-adic GL2(𝒟)

2007 ◽  
Vol 59 (5) ◽  
pp. 1050-1068 ◽  
Author(s):  
A. Raghuram
Keyword(s):  
Irreducible Representation ◽  
Division Algebra ◽  
Local Field ◽  
Joint Work ◽  
Invariant Distributions ◽  
Multiplicity One ◽  
Kirillov Theory ◽  
Multiplicity Free ◽  
Diagonal Subgroup ◽  
Quaternion Division Algebra

AbstractLet be a division algebra over a nonarchimedean local field. Given an irreducible representation π of GL2(), we describe its restriction to the diagonal subgroup × . The description is in terms of the structure of the twisted Jacquet module of the representation π. The proof involves Kirillov theory that we have developed earlier in joint work with Dipendra Prasad. The main result on restriction also shows that π is × -distinguished if and only if π admits a Shalika model. We further prove that if is a quaternion division algebra then the twisted Jacquetmodule is multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to × in the quaternionic case.

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 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.

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