Uniform bounds on the Harish-Chandra characters

Let $\mathbf{G}$ be a connected reductive algebraic group over a $p$-adic local field $F$. In this paper we study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular element $\gamma$ of $\mathbf{G}(F)$ as $\pi$ varies among supercuspidal representations of $\mathbf{G}(F)$. Kim, Shin and Templier conjectured that $\frac{\theta_{\pi}(\gamma)}{{\rm deg}(\pi)}$ tends to $0$ when $\pi$ runs over irreducible supercuspidal representations of $\textbf{G}(F)$ with unitary central character and the formal degree of $\pi$ tends to infinity. For $\textbf{G}$ semisimple we prove that the trace character is uniformly bounded on $\gamma$ under the assumption, which is expected to hold true for every $\textbf{G} (F)$, that all irreducible supercuspidal representations of $\textbf{G}(F)$ are compactly induced from an open compact modulo center subgroup. Moreover, we give an explicit upper bound in the case of $\gamma$ ellitpic.Comment: Added explicit bound in the elliptic cas

Simultaneous pp-orderings and minimising volumes in number fields

In the paper "On the interpolation of integer-valued polynomials" (Journal of Number Theory 133 (2013), pp. 4224--4232.) V. Volkov and F. Petrov consider the problem of existence of the so-called $n$-universal sets (related to simultaneous $p$-orderings of Bhargava) in the ring of Gaussian integers. We extend their results to arbitrary imaginary quadratic number fields and prove an existence theorem that provides a strong counterexample to a conjecture of Volkov-Petrov on minimal cardinality of $n$-universal sets. Along the way, we discover a link with Euler-Kronecker constants and prove a lower bound on Euler-Kronecker constants which is of the same order of magnitude as the one obtained by Ihara.Comment: new version, substantial corrections in section 6, will appear in Journal of Number Theor

Cuspidal types on GLp(OF)\textrm{GL}_{p}(\mathcal{O}_{F})

Let $F$ be a non-Archimedean local field and let $\mathcal{O}_{F}$ be its ring of integers. We give a description of orbits of cuspidal types on $\mathrm{GL}_{p}( \mathcal{O}_{F})$, with $p$ prime. We determine which of them are regular representations and we provide an example which shows that an orbit of a representation does not always determine whether it is a cuspidal type or not.Comment: 28 page

Regular representations of GL _n(O) and the inertial Langlands correspondence

This thesis is divided into two parts. The first one comes from the representation theory of reductive $\mathfrak{p}$-adic groups. The main motivation behind this part of the thesis is to find new explicit information and invariants of the types in general linear groups. Let $F$ be a non-Archimedean local field and let $\mathcal{O}_{F}$ be its ring of integers. We give an explicit description of cuspidal types on \GL _{p}(\mathcal{O}_{F}), with $p$ prime, in terms of orbits. We determine which of them are regular representations and we provide an example which shows that an orbit of a representation does not always determine whether it is a cuspidal type or not. At the same time we prove that a cuspidal type for a representation $\pi$ of $GL_{p}(F)$ is regular if and only if the normalised level of $\pi$ is equal to $m$ or $m -\frac{1}{p}$ for $m \in \mathbb{Z}$. The second part of the thesis comes from the theory of integer-valued polynomials and simultaneous $\mathfrak{p}$-orderings. This is a joint work with Mikołaj Frączyk. The notion of simultaneous $\mathfrak{p}$-ordering was introduced by Bhargava in his early work on integer-valued polynomials. Let $k$ be a number field and let $\mathcal{O}_{k}$ be its ring of integers. Roughly speaking a simultaneous $\mathfrak{p}$-ordering is a sequence of elements from $\mathcal{O}_{k}$ which is equidistributed modulo every power of every prime ideal in $\mathcal{O}_{k}$ as well as possible. Bhargava asked which subsets of Dedekind domains admit simultaneous $\mathfrak{p}$-ordering. Together with Mikołaj Frączyk we proved that the only number field $k$ with $\mathcal{O}_{k}$ admitting a simultaneous $\mathfrak{p}$-ordering is $\mathbb{Q}$

Représentations régulières de GLn( O) et la correspondance de Langlands inertielle

Cette thèse contient deux parties. La première porte sur la théorie des représentations des groupes p-adiques. Le but est de trouver de nouvelles informations et de nouveaux invariants des types cuspidaux de groupes linéaires généraux. Soit F un corps local non archimédien et soit OF son anneau des entiers. Nous décrivons les types cuspidaux sur GLp(OF ) (où p est un nombre premier) en termes d’orbites. Nous déterminons quels types cuspidaux sont réguliers et donnons un exemple qui montre que l’orbite de la représentation ne suffit pas à déterminer si la représentation est un type cuspidal ou non. Nous montrons qu’un type cuspidal pour une représentation π de GLp(F) est régulier si et seulement si le niveau normalisé de π est égal à m ou m − 1 p pour un certain m ∈ Z. La deuxième partie porte sur les polynômes à valeurs entières, les p-rangements simultanés (au sens de Bhargava) et l’équidistribution dans les corps des nombres. C’est un projet joint avec Mikołaj Frączyk. La notion de p-rangement provient des travaux de Bhargava sur les polynômes à valeurs entières. Soit k un corps de nombres et soit Ok son anneau des entiers. Une suite d’éléments de Ok est un p-rangement simultané si elle est équidistribuée modulo tous les idéaux premières de Ok du mieux possible. Nous prouvons que le seul corps de nombres k tel que Ok admette des p-rangements simultanés est Q.This thesis is divided into two parts. The first one comes from the representation theory of reductive p-adic groups. The main motivation behind this part of the thesis is to find new explicit information and invariants of the types in general linear groups. Let F be a nonArchimedean local field and let OF be its ring of integers. We give an explicit description of cuspidal types on GLp(OF ), with p prime, in terms of orbits. We determine which of them are regular representations and we provide an example which shows that an orbit of a representation does not always determine whether it is a cuspidal type or not. At the same time we prove that a cuspidal type for a representation π of GLp(F) is regular if and only if the normalised level of π is equal to m or m − 1 p for m ∈ Z. The second part of the thesis comes from the theory of integer-valued polynomials and simultaneous p-orderings. This is a joint work with Mikołaj Frączyk. The notion of simultaneous p-ordering was introduced by Bhargava in his early work on integer-valued polynomials. Let k be a number field and let Ok be its ring of integers. Roughly speaking a simultaneous p-ordering is a sequence of elements from Ok which is equidistributed modulo every power of every prime ideal in Ok as well as possible. Bhargava asked which subsets of Dedekind domains admit simultaneous p-ordering. Together with Mikołaj Frączyk we proved that the only number field k with Ok admitting a simultaneous p-ordering is Q

On representations of GLn(Qp)GL_n(\mathbb{Q}_p) in characteristic pp.

In the paper, we will show the classification of irreducible smooth representations of $GL_n(\mathbb{Q}_{p})$ over \overline{\mathbb{F}}_p} following the work of Florian Herzig. The methods we will use in the thesis include changing weights, and the notions of Hecke algebras and Satake isomorphism.In the paper, we will show the classification of irreducible smooth representations of $GL_n(\\mathbb{Q}_{p})$ over \\overline{\\mathbb{F}}_p} following the work of Florian Herzig. The methods we will use in the thesis include changing weights, and the notions of Hecke algebras and Satake isomorphism

Représentations régulières de GLn( O) et la correspondance de Langlands inertielle

This thesis is divided into two parts. The first one comes from the representation theory of reductive p-adic groups. The main motivation behind this part of the thesis is to find new explicit information and invariants of the types in general linear groups. Let F be a nonArchimedean local field and let OF be its ring of integers. We give an explicit description of cuspidal types on GLp(OF ), with p prime, in terms of orbits. We determine which of them are regular representations and we provide an example which shows that an orbit of a representation does not always determine whether it is a cuspidal type or not. At the same time we prove that a cuspidal type for a representation π of GLp(F) is regular if and only if the normalised level of π is equal to m or m − 1 p for m ∈ Z. The second part of the thesis comes from the theory of integer-valued polynomials and simultaneous p-orderings. This is a joint work with Mikołaj Frączyk. The notion of simultaneous p-ordering was introduced by Bhargava in his early work on integer-valued polynomials. Let k be a number field and let Ok be its ring of integers. Roughly speaking a simultaneous p-ordering is a sequence of elements from Ok which is equidistributed modulo every power of every prime ideal in Ok as well as possible. Bhargava asked which subsets of Dedekind domains admit simultaneous p-ordering. Together with Mikołaj Frączyk we proved that the only number field k with Ok admitting a simultaneous p-ordering is Q.Cette thèse contient deux parties. La première porte sur la théorie des représentations des groupes p-adiques. Le but est de trouver de nouvelles informations et de nouveaux invariants des types cuspidaux de groupes linéaires généraux. Soit F un corps local non archimédien et soit OF son anneau des entiers. Nous décrivons les types cuspidaux sur GLp(OF ) (où p est un nombre premier) en termes d’orbites. Nous déterminons quels types cuspidaux sont réguliers et donnons un exemple qui montre que l’orbite de la représentation ne suffit pas à déterminer si la représentation est un type cuspidal ou non. Nous montrons qu’un type cuspidal pour une représentation π de GLp(F) est régulier si et seulement si le niveau normalisé de π est égal à m ou m − 1 p pour un certain m ∈ Z. La deuxième partie porte sur les polynômes à valeurs entières, les p-rangements simultanés (au sens de Bhargava) et l’équidistribution dans les corps des nombres. C’est un projet joint avec Mikołaj Frączyk. La notion de p-rangement provient des travaux de Bhargava sur les polynômes à valeurs entières. Soit k un corps de nombres et soit Ok son anneau des entiers. Une suite d’éléments de Ok est un p-rangement simultané si elle est équidistribuée modulo tous les idéaux premières de Ok du mieux possible. Nous prouvons que le seul corps de nombres k tel que Ok admette des p-rangements simultanés est Q