Towards the Jacquet conjecture on the Local Converse Problem for $p$-adic $\mathrm {GL}_n$

Dihua Jiang; Chufeng Nien; Shaun Stevens

Displaying similar documents to “Towards the Jacquet conjecture on the Local Converse Problem for $p$ -adic ${GL}_{n}$ ”

Locally analytic vectors of unitary principal series of ${GL}_{2} (ℚ_{p})$

Ruochuan Liu, Bingyong Xie, Yuancao Zhang (2012)

Annales scientifiques de l'École Normale Supérieure

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The $p$ -adic local Langlands correspondence for ${GL}_{2} (ℚ_{p})$ attaches to any $2$ -dimensional irreducible $p$ -adic representation $V$ of $G_{ℚ_{p}}$ an admissible unitary representation $Π (V)$ of ${GL}_{2} (ℚ_{p})$ . The unitary principal series of ${GL}_{2} (ℚ_{p})$ are those $Π (V)$ corresponding to trianguline representations. In this article, for $p > 2$ , using the machinery of Colmez, we determine the space of locally analytic vectors $Π {(V)}_{an}$ for all non-exceptional unitary principal series $Π (V)$ of ${GL}_{2} (ℚ_{p})$ by proving a conjecture of Emerton.

Asymptotic values of modular multiplicities for ${GL}_{2}$

Sandra Rozensztajn (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the irreducible constituents of the reduction modulo $p$ of irreducible algebraic representations $V$ of the group ${Res}_{K / ℚ_{p}} {GL}_{2}$ for $K$ a finite extension of $ℚ_{p}$ . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of $V$ and the central character of its reduction modulo $p$ . As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

Invariance of the parity conjecture for $p$ -Selmer groups of elliptic curves in a $D_{2 p^{n}}$ -extension

Thomas de La Rochefoucauld (2011)

Bulletin de la Société Mathématique de France

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We show a $p$ -parity result in a $D_{2 p^{n}}$ -extension of number fields $L / K$ ( $p \geq 5$ ) for the twist $1 \oplus η \oplus τ$ : $W (E / K, 1 \oplus η \oplus τ) = {(- 1)}^{〈1 \oplus η \oplus τ, X_{p} (E / L)〉}$ , where $E$ is an elliptic curve over $K$ , $η$ and $τ$ are respectively the quadratic character and an irreductible representation of degree $2$ of $Gal (L / K) = D_{2 p^{n}}$ , and $X_{p} (E / L)$ is the $p$ -Selmer group. The main novelty is that we use a congruence result between $ε_{0}$ -factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$ -parity conjecture...

Dual Blobs and Plancherel Formulas

Ju-Lee Kim (2004)

Bulletin de la Société Mathématique de France

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Let $k$ be a $p$ -adic field. Let $G$ be the group of $k$ -rational points of a connected reductive group $𝖦$ defined over $k$ , and let $𝔤$ be its Lie algebra. Under certain hypotheses on $𝖦$ and $k$ , wethe tempered dual $\hat{G}$ of $G$ via the Plancherel formula on $𝔤$ , using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on $𝔤$ and $G$ . As a consequence, we prove that any tempered representation contains a good minimal $𝖪$ -type; we extend this result to irreducible...

A new characterization for the simple group $PSL (2, p^{2})$ by order and some character degrees

Behrooz Khosravi, Behnam Khosravi, Bahman Khosravi, Zahra Momen (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $| PSL (2, p^{2}) |$ such that $G$ has an irreducible character of degree $p^{2}$ and we know that $G$ has no irreducible character $θ$ such that $2 p ∣ θ (1)$ , then $G$ is isomorphic to $PSL (2, p^{2})$ . As a consequence of our result we prove that $PSL (2, p^{2})$ is uniquely determined by the structure of its complex group algebra.

Generalized symmetry classes of tensors

Gholamreza Rafatneshan, Yousef Zamani (2020)

Czechoslovak Mathematical Journal

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Let $V$ be a unitary space. For an arbitrary subgroup $G$ of the full symmetric group $S_{m}$ and an arbitrary irreducible unitary representation $Λ$ of $G$ , we study the generalized symmetry class of tensors over $V$ associated with $G$ and $Λ$ . Some important properties of this vector space are investigated.

Thompson’s conjecture for the alternating group of degree $2 p$ and $2 p + 1$

Azam Babai, Ali Mahmoudifar (2017)

Czechoslovak Mathematical Journal

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For a finite group $G$ denote by $N (G)$ the set of conjugacy class sizes of $G$ . In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N (G) = N (L)$ , then $G ≅ L$ . We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z (G) = 1$ and $N (G) = N (A_{i})$ is necessarily isomorphic to $A_{i}$ , where $i \in {2 p, 2 p + 1}$ .

Representations of the general linear group over symmetry classes of polynomials

Yousef Zamani, Mahin Ranjbari (2018)

Czechoslovak Mathematical Journal

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Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_{1}, ..., x_{m}$ . Suppose $G$ is a subgroup of $S_{m}$ , and $χ$ is an irreducible character of $G$ . Let $H_{d} (G, χ)$ be the symmetry class of polynomials of degree $d$ with respect to $G$ and $χ$ . For any linear operator $T$ acting on $V$ , there is a (unique) induced operator $K_{χ} (T) \in End (H_{d} (G, χ))$ acting on symmetrized decomposable polynomials by $K_{χ} (T) (f_{1} * f_{2} * ... * f_{d}) = T f_{1} * T f_{2} * ... * T f_{d} .$ In this paper, we show that the representation $T \mapsto K_{χ} (T)$ of the general linear group $G L (V)$ is equivalent to the direct sum of $χ (1)$ copies...

The centralizer of a classical group and Bruhat-Tits buildings

Daniel Skodlerack (2013)

Annales de l’institut Fourier

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Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G .$ We prove that the Bruhat-Tits building $𝔅^{1} (H)$ of $H$ can be affinely and $G$ -equivariantly embedded in the Bruhat-Tits building $𝔅^{1} (G)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^{'}$ be maps from $𝔅^{1} (H)$ to $𝔅^{1} (G)$ which preserve the Moy–Prasad filtrations....

On the infinite fern of Galois representations of unitary type

Gaëtan Chenevier (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $E$ be a CM number field, $p$ an odd prime totally split in $E$ , and let $X$ be the $p$ -adic analytic space parameterizing the isomorphism classes of $3$ -dimensional semisimple $p$ -adic representations of $Gal (\bar{E} / E)$ satisfying a selfduality condition “of type $U (3)$ ”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3 [E : ℚ]$ . As important steps, and in any rank, we prove that any...

Results related to Huppert’s $ρ$ - $σ$ conjecture

Xia Xu, Yong Yang (2023)

Czechoslovak Mathematical Journal

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We improve a few results related to Huppert’s $ρ$ - $σ$ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups.

On sharp characters of type ${- 1, 0, 2}$

Alireza Abdollahi, Javad Bagherian, Mahdi Ebrahimi, Maryam Khatami, Zahra Shahbazi, Reza Sobhani (2022)

Czechoslovak Mathematical Journal

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For a complex character $χ$ of a finite group $G$ , it is known that the product $sh (χ) = \prod_{l \in L (χ)} (χ (1) - l)$ is a multiple of $| G |$ , where $L (χ)$ is the image of $χ$ on $G - {1}$ . The character $χ$ is said to be a sharp character of type $L$ if $L = L (χ)$ and $sh (χ) = | G |$ . If the principal character of $G$ is not an irreducible constituent of $χ$ , then the character $χ$ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups $G$ with normalized sharp characters of type ${- 1, 0, 2}$ . Here we prove that such a group with nontrivial...

On a conjecture of Dekking : The sum of digits of even numbers

Iurie Boreico, Daniel El-Baz, Thomas Stoll (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $q \geq 2$ and denote by $s_{q}$ the sum-of-digits function in base $q$ . For $j = 0, 1, \dots, q - 1$ consider $# {0 \leq n < N : s_{q} (2 n) \equiv j (mod q)} .$ In 1983, F. M. Dekking conjectured that this quantity is greater than $N / q$ and, respectively, less than $N / q$ for infinitely many $N$ , thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

Irreducibility of automorphic Galois representations of $G L (n)$ , $n$ at most $5$

Frank Calegari, Toby Gee (2013)

Annales de l’institut Fourier

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Let $π$ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of ${GL}_{n} (𝔸_{F})$ , where $F$ is a totally real field and $n$ is at most $5$ . We show that for all primes $l$ , the $l$ -adic Galois representations associated to $π$ are irreducible, and for all but finitely many primes $l$ , the mod $l$ Galois representations associated to $π$ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$ -adic representations are independent of $l$ .

Displaying similar documents to “Towards the Jacquet conjecture on the Local Converse Problem for p -adic GL n ”

Displaying similar documents to “Towards the Jacquet conjecture on the Local Converse Problem for $p$ -adic ${GL}_{n}$ ”