Quasi-semi-stable representations

Xavier Caruso; Tong Liu

Displaying similar documents to “Quasi-semi-stable representations”

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 $l$ .

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...

Arithmetic Properties of Generalized Rikuna Polynomials

Z. Chonoles, J. Cullinan, H. Hausman, A.M. Pacelli, S. Pegado, F. Wei (2014)

Publications mathématiques de Besançon

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Fix an integer $ℓ \geq 3$ . Rikuna introduced a polynomial $r (x, t)$ defined over a function field $K (t)$ whose Galois group is cyclic of order $ℓ$ , where $K$ satisfies some mild hypotheses. In this paper we define the family of ${r_{n} (x, t)}_{n \geq 1}$ of degree $ℓ^{n}$ . The $r_{n} (x, t)$ are constructed iteratively from the $r (x, t)$ . We compute the Galois groups of the $r_{n} (x, t)$ for odd $ℓ$ over an arbitrary base field and give applications to arithmetic dynamical systems.

Fields of moduli of three-point $G$ -covers with cyclic $p$ -Sylow, II

Andrew Obus (2013)

Journal de Théorie des Nombres de Bordeaux

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We continue the examination of the stable reduction and fields of moduli of $G$ -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $(0, p)$ , where $G$ has a $p$ -Sylow subgroup $P$ of order $p^{n}$ . Suppose further that the normalizer of $P$ acts on $P$ via an involution. Under mild assumptions, if $f : Y \to ℙ^{1}$ is a three-point $G$ -Galois cover defined over $\bar{ℚ}$ , then the $n$ th higher ramification groups above $p$ for the upper numbering of the (Galois closure of...

Counting discriminants of number fields

Henri Cohen, Francisco Diaz y Diaz, Michel Olivier (2006)

Journal de Théorie des Nombres de Bordeaux

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For each transitive permutation group $G$ on $n$ letters with $n \leq 4$ , we give without proof results, conjectures, and numerical computations on discriminants of number fields $L$ of degree $n$ over $ℚ$ such that the Galois group of the Galois closure of $L$ is isomorphic to $G$ .

When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Stéphane R. Louboutin (2020)

Czechoslovak Mathematical Journal

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Let $ε$ be an algebraic unit of the degree $n \geq 3$ . Assume that the extension $ℚ (ε) / ℚ$ is Galois. We would like to determine when the order $ℤ [ε]$ of $ℚ (ε)$ is $Gal (ℚ (ε) / ℚ)$ -invariant, i.e. when the $n$ complex conjugates $ε_{1}, \dots, ε_{n}$ of $ε$ are in $ℤ [ε]$ , which amounts to asking that $ℤ [ε_{1}, \dots, ε_{n}] = ℤ [ε]$ , i.e., that these two orders of $ℚ (ε)$ have the same discriminant. This problem has been solved only for $n = 3$ by using an explicit formula for the discriminant of the order $ℤ [ε_{1}, ε_{2}, ε_{3}]$ . However, there is no known similar formula for $n > 3$ . In the present paper, we put forward and...

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...

On classifying Laguerre polynomials which have Galois group the alternating group

Pradipto Banerjee, Michael Filaseta, Carrie E. Finch, J. Russell Leidy (2013)

Journal de Théorie des Nombres de Bordeaux

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We show that the discriminant of the generalized Laguerre polynomial $L_{n}^{(α)} (x)$ is a non-zero square for some integer pair $(n, α)$ , with $n \geq 1$ , if and only if $(n, α)$ belongs to one of $30$ explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of $L_{n}^{(α)} (x)$ over $ℚ$ is the alternating group $A_{n}$ . For example, we establish that for all but finitely many positive integers $n \equiv 2 (mod 4)$ , the only $α$ for which the Galois group of $L_{n}^{(α)} (x)$ over $ℚ$ is $A_{n}$ is...

Piecewise hereditary algebras under field extensions

Jie Li (2021)

Czechoslovak Mathematical Journal

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Let $A$ be a finite-dimensional $k$ -algebra and $K / k$ be a finite separable field extension. We prove that $A$ is derived equivalent to a hereditary algebra if and only if so is $A \otimes_{k} K$ .