Séminaire de Théorie des Nombres de Nancy-Metz - Institut Élie Cartan de Lorraine

Exposés à venir

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Multiplicative orders mod $p$

Catégorie d'évènement : Séminaire de Théorie des Nombres de Nancy-Metz Date/heure : 8 avril 2021 14:30-15:30 Lieu : Salle de séminaire de Théorie des Nombres virtuelle Oratrice ou orateur : Paul Pollack (University of Georgia) Résumé :

I will survey what is known about the distribution of the orders of integers mod $p$, as $p$ varies. Particular attention will be paid to problems of the following sort: For fixed $a$ and $b$, how do the order of $a$ mod $p$ and the order of $b$ mod $p$ compare, as $p$ varies? The proofs will draw from the elementary, algebraic, and analytic strands of number theory. (So hopefully something for everyone!)

Modular zeros in the character table of the symmetric group

Catégorie d'évènement : Séminaire de Théorie des Nombres de Nancy-Metz Date/heure : 1 avril 2021 15:30-16:30 Lieu : Salle de séminaire de Théorie des Nombres virtuelle Oratrice ou orateur : Sarah Peluse (IAS/Princeton) Résumé :

In 2017, Miller conjectured, based on computational evidence, that for any fixed prime $p$ the density of entries in the character table of $S_n$ that are divisible by $p$ goes to $1$ as $n$ goes to infinity. I’ll describe a proof of this conjecture, which is joint work with K. Soundararajan. I will also discuss the (still open) problem of determining the asymptotic density of zeros in the character table of $S_n$, where it is not even clear from computational data what one should expect.

An approximate form of Artin's holomorphy conjecture and nonvanishing of Artin L-functions

Catégorie d'évènement : Séminaire de Théorie des Nombres de Nancy-Metz Date/heure : 18 mars 2021 15:30-16:30 Lieu : Salle de séminaire de Théorie des Nombres virtuelle Oratrice ou orateur : Asif Zaman (University of Toronto) Résumé :

Let $k$ be a number field and $G$ be a finite group, and let $\mathfrak{F}_{k}^{G}$ be a family of number fields $K$ such that $K/k$ is normal with Galois group isomorphic to $G$. Together with Robert Lemke Oliver and Jesse Thorner, we prove for many families that for almost all $K \in \mathfrak{F}_k^G$, all of the $L$-functions associated to Artin representations whose kernel does not contain a fixed normal subgroup are holomorphic and non-vanishing in a wide region.

These results have several arithmetic applications. For example, we prove a strong effective prime ideal theorem that holds for almost all fields in several natural large degree families, including the family of degree $n$ $S_n$-extensions for any $n \geq 2$ and the family of prime degree $p$ extensions (with any Galois structure) for any prime $p \geq 2$. I will discuss this result, describe the main ideas of the proof, and share some applications to bounds on $\ell$-torsion subgroups of class groups, to the extremal order of class numbers, and to the subconvexity problem for Dedekind zeta functions.

On computing $L’/L(1,\chi)$ and related problems

Catégorie d'évènement : Séminaire de Théorie des Nombres de Nancy-Metz Date/heure : 11 mars 2021 14:30-15:30 Lieu : Salle de séminaire de Théorie des Nombres virtuelle Oratrice ou orateur : Alessandro Languasco Résumé :

We first describe an efficient algorithm to compute
$L’/L(1,\chi)$, where $\chi$ is a non-principal Dirichlet character
mod q, and q is an odd prime. We then discuss
some results on the distribution of
$m_q := \min_{\chi\ne \chi_0} \vert L’/L(1,\chi) \vert $
and about the Euler-Kronecker constants for cyclotomic fields.

The distribution of random polynomials with multiplicative coefficients

Catégorie d'évènement : Analyse et théorie des nombres Date/heure : 18 février 2021 14:30-15:30 Lieu : Oratrice ou orateur : Brad Rodgers Résumé :

A classic paper of Salem and Zygmund investigates the distribution of trigonometric polynomials whose coefficients are chosen randomly (say +1 or -1 with equal probability) and independently. Salem and Zygmund characterized the typical distribution of such polynomials (gaussian) and the typical magnitude of their sup-norms (a degree N polynomial typically has sup-norm of size $\sqrt{N \log N}$ for large N). In this talk we will explore what happens when a weak dependence is introduced between coefficients of the polynomials; namely we consider polynomials with coefficients given by random multiplicative functions. We consider analogues of Salem and Zygmund’s results, exploring similarities and some differences.

Special attention will be given to a beautiful point-counting argument introduced by Vaughan and Wooley which ends up being useful.

This is joint work with Jacques Benatar and Alon Nishry.

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Exposés à venir

Archives

Multiplicative orders mod $p$

Modular zeros in the character table of the symmetric group

An approximate form of Artin's holomorphy conjecture and nonvanishing of Artin L-functions

On computing $L’/L(1,\chi)$ and related problems

The distribution of random polynomials with multiplicative coefficients

Non-vanishing of cubic L-functions

Comportement aléatoire local des suites réelles: résultats métriques, énergie additive et inégalités diophantiennes

Théorèmes d'Erdös-Wintner effectifs

Sur la répartition jointe de la représentation d'Ostrowski dans les classes de résidus

Soutenance de thèse : Lois gaussiennes inverses généralisées, lois de Kummer et méthode de Stein