Number Theory Days in Lille

8 juil. 2019, 08:00 → 11 juil. 2019, 16:45 Europe/Paris

Salle de réunion (Université de Lille)

Overview

The purpose of the conference is to bring together well renowned experts in algebra and number theory in order to focus on the latest advances in the fields where the laboratories of Lille and its region are particularly involved: noncommutative algebra, quadratic forms, arithmetic geometry, Galois theory, and analytic number theory.

Organisers

Niels Borne (Lille)

Julien Hauseux (Lille)

Ahmed Laghribi (Artois)

Scientific committee

Gautami Bhowmik (Lille)	Baptiste Calmès (Artois)	Raf Cluckers (Lille / KU Leuven)
Pierre Dèbes (Lille)	Mladen Dimitrov (Lille)	André Leroy (Artois)
	Nicole Raulf (Lille)

Invited speakers

Roberto Aravire (Universidad Arturo Prat)	Daniel Barrera (Universidad de Santiago de Chile)
Karim Johannes Becher (Universiteit Antwerpen)	Victoria Cantoral-Farfan (ICTP)
Philip Dittmann (KU Leuven)	Alberto Facchini (Università di Padova)
Walter Ferrer (Universidad de la República)	Gergely Harcos (Alfréd Rényi Institute of Mathematics)
Dimitar Jetchev (EPFL)	Mahesh Kakde (King's College London)
Joachim König (KAIST)	Matilde Lalín (Université de Montréal)
Youness Lamzouri (Université de Lorraine / York University)	Jaclyn Lang (Université Paris 13)
Paul Nelson (ETH Zurich)	Tom Sanders (University of Oxford)
Blas Torrecillas (Universidad de Almería)

Sponsors

lun. 8 juillet

09:00 → 16:30 Analytic-additive number theory

09:00 Coffee and pastries

09:45 A Minkowski-type result for linearly independent subsets of ideal lattices

We estimate, in a number field, the maximal number of linearly independent elements with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices. Our arguments combine group theory, ramification theory, and the geometry of numbers. This is joint work with Mikołaj Frączyk and Péter Maga.

Orateur: Gergely Harcos (Alfréd Rényi Institute of Mathematics)

10:45 Break

11:00 On the distribution of the maximum of partial sums of exponential sums

In analogy with multiplicative character sums, we investigate the distribution of the maximum of partial sums of various families of exponential sums. We obtain precise estimates on the distribution function in a large uniform range, in the case where the Fourier transforms of these exponential sums are real valued, and satisfy some "natural" hypotheses. Important examples include Birch sums and Kloostermann sums. The proof uses a blend of analytic and probabilistic techniques together with deep tools from algebraic geometry. As an application, we exhibit large values of partial sums of these exponential sums, which we believe are best possible. This is a joint work with Pascal Autissier and Dante Bonolis.

Orateur: Youness Lamzouri (Université de Lorraine / York University)

12:00 Lunch break

14:00 The mean value of cubic L-functions over function fields

We present results about the first moment of L-functions associated to cubic characters over $\mathbb{F}_q(T)$ when q is congruent to 1 modulo 3. The case of number fields was considered in previous work, but never for the full family of cubic twists over a field containing the third roots of unity. We will explain how to obtain an asymptotic formula with a main term, which relies on using results from the theory of metaplectic Eisenstein series about cancellation in averages of cubic Gauss sums over functions fields. We will also discuss the case q congruent to 2 modulo 3.

Orateur: Matilde Lalín (Université de Montréal)

15:00 Coffee break

15:30 Bootstrapping partition regularity of linear systems

Suppose that $A$ is a $k \times d$ matrix of integers such that fir any $r$ there is some $N$ such that any $r$-colouring of $\{1,\dots,N\}$ contains a monochromatic solution to $A$, meaning there is a colour class $C$ and $x \in C^d$ such that $Ax=0$. Not all matrices $A$ have this property (consider, for example, when all the entries of $A$ are positive), but when they do they are called partition regular. In this talk we consider what bounds can be given on $N$ in terms of $r$ (and $A$) when $A$ is partition regular.

Orateur: Tom Sanders (University of Oxford)

mar. 9 juillet

09:00 → 16:30 Galois representations and modular forms

09:00 Coffee and pastries

09:45 Applications of the orbit method to the analysis of automorphic forms

I will discuss joint work with Akshay Venkatesh in which we use microlocalized test vectors (inspired by the orbit method) and Ratner theory to study mean values of L-functions on Gross--Prasad pairs. I will also indicate some further applications of these methods, such as to the quantum variance problem for Hecke--Maass eigenforms.

Orateur: Paul Nelson (ETH Zurich)

10:45 Break

11:00 Birch and Swinnerton-Dyer Formula for modular forms of arbitrary weight in the cases of analytic ranks 0 and 1

In this talk, I will report on recent results on the computation of the p-part of the leading term of the L-function of a modular form of arbitrary weight at the central point in the cases when the order of vanishing is at most one. Unlike the classical case of weight 2 modular forms, qualitatively different arguments are needed in the higher-weight case. After explaining the difference, I will indicate how one can use level-raising and (non-ordinary) p-adic deformations together with some of the arguments in weight 2 to obtain results in the case of general weights.
This is joint work with Chris Skinner and Xin Wan.

Orateur: Dimitar Jetchev (EPFL)

12:00 Lunch break

14:00 On a conjecture of Gross and explicit formula for Gross-Stark units

In this talk I will report on my joint work in progress with Samit Dasgupta on the tower of fields conjecture first formulated by Gross. This proves a conjecture of Dasgupta on explicit p-adic analytic formulae for Gross-Stark units. These units, when considered for all primes of a totally real number field F, generate the maximal abelian CM extension of F and therefore our work can be considered as giving a p-adic analytic solution to Hilbert’s 12th problem.

Orateur: Mahesh Kakde (King's College London)

15:00 Coffee break

15:30 Overconvergent cohomology and automorphic p-adic L-functions

p-adic L-functions attached to automorphic representations and p-adic families of them, provide powerful tools to attack important problems such as Birch-Swinnerton-Dyer and Bloch-Kato conjetures. However, they are hard to construct and in fact beyond the case GL(2) the theory is poorly understood.

In this talk I will describe an approach based on the study of the overconvergent cohomology of locally symmetric spaces. This approach was introduced by G. Stevens in the nineties and the most general constructions available for GL(2) were based on it. Then I will describe an ongoing joint work with M. Dimitrov and C. Williams in which we construct p-adic L-functions for certain cuspidal automorphic representations of GL(2n) by the use of convergent cohomology. This construction extends previous results of Gehrmann/Dimitrov-Januszewski-Raghuram to the non-ordinary setting and allows variation in p-adic families.

Orateur: Daniel Barrera (Universidad de Santiago de Chile)

mer. 10 juillet

09:00 → 12:00 Quadratic forms

09:00 Coffee and pastries

09:45 Quadratic forms and diophantine sets

The interplay between valuations and certain geometrically rational varieties, in particular quadrics, has turned out to be very fruitful for proving that certain subsets of fields are existentially definable or diophantine. In particular, this has been used by J. Koenigsmann to prove that Q\Z is diophantine in Q. His proof combines several ingredients from classical number theory, involving in particular the Hasse-Minkowski local-global principle for quadratic forms. In my talk I want to highlight some ingredients of proofs for showing that certain subsets of fields are diophantine and some interesting questions for quadratic forms arising from this context.

Orateur: Karim Johannes Becher (Universiteit Antwerpen)

10:45 Break

11:00 Quadratic and Differential Forms over fields of characteristic 2

In this talk $F$ denotes a field of characteristic $2$, $W_{q}(F)$ the Witt of nonsingular quadratic forms over $F$, $W(F)$ the Witt ring of regular symmetric bilinear forms over $F$. For any integer $m\geq0$, we denote by $I_{q}^{m+1}(F)$ the group $I^{m}F\otimes W_{q}(F)$, where $I^{m}F$ \ is the $m$-th power of the fundamental ideal $IF$ of $W(F)$, and $\otimes$ is the module action of $W(F)$ on\ $W_{q}(F)$.

Given a field extension $K$ of $F$, we have two homomorphisms $i_{K}:W_{q}(F)\longrightarrow W_{q}(K)$ and $j_{K}:W(F)\longrightarrow W(K)$ induced by the inclusion $F\subset K$. A natural question that arises is to compute the kernels of these homomorphisms. This is an outstanding problem in the theory of quadratic forms.

A way to study the above problems is based on differential forms where we use a celebrated result of K. Kato which gives connections between quadratic forms and differential forms, and, by this way, obtain results in terms of graded Witt groups.

Following this approach we will provide examples who illustrate the use of tools.

Orateur: Roberto Aravire (Universidad Arturo Prat)

12:00 → 14:00 Lunch break

14:00 → 17:45 Noncommutative algebra

14:00 Pretorsion theories in general categories

We will present a notion of (pre)torsion theory in general categories and two interesting examples of such pretorsion theories. Torsion theories in arbitrary categories have been studied by Grandis, Janelidze, Márki and several others. Our main examples will be in the category of preordered sets and the category of finite algebras with one operation, unary, and no axioms (i.e., the category of all mappings $f\colon X\to X$, where $X$ is a finite set). Our results appear in joint papers with Carmelo Antonio Finocchiaro, Marino Gran and Leila Heidari Zadeh.

Orateur: Alberto Facchini (Università di Padova)

15:00 Coffee break

15:30 Galois theory for cowreaths

The Galois theory for monoidal cowreaths is developed. Cleft cowreaths are introduced in this context and its relation with the normal basis property investigated. The connection of this class of cowreath with some wreath algebra structures is obtained. Finally, several applications to quasi-Hopf algebras will be discussed. This is a joint work with D. Bulacu.

Orateur: Blas Torrecillas (Universidad de Almería)

16:30 Break

16:45 Hopf Ore extensions and the antipode

In this talk we deal with Hopf Ore extensions, the role they play in the classification of low dimensional Hopf algebras, and the property of a Hopf algebra to be almost involutive (meaning that the square of the antipode has a square root that is an automorphism of Hopf algebras).

Orateur: Walter Ferrer (Universidad de la República)

jeu. 11 juillet

09:00 → 09:45 Coffee and pastries

09:45 → 10:45 Galois representations and modular forms

09:45 Image of two-dimensional pseudorepresentations

There is a general philosophy that the image of a Galois representation should be as large as possible, subject to its symmetries. This can be seen in Serre's open image theorem for non-CM elliptic curves, Ribet and Momose's work on Galois representations attached to modular forms, and recent work of the speak and Conti-Iovita-Tilouine on Galois representations attached to p-adic families of modular forms. Recently, Bellaïche developed a way to measure the image of an arbitrary pseudorepresentations taking values in a local ring A. Under the assumptions that A is a domain and the residual representation is not too degenerate, we explain how the symmetries of such a pseudorepresentation are reflected in its image. This is joint work with Andrea Conti and Anna Medvedovsky.

Orateur: Jaclyn Lang (Université Paris 13)

10:45 → 11:00 Break

11:00 → 16:00 Arithmetic geometry and Galois theory

11:00 Density of specialization sets and Hasse principle in families of twisted Galois covers

We discuss results on the structure of the set of all specializations of a Galois cover f:X->P^1 with group G over a number field k. Hilbert's irreducibility theorem yields that this set contains infinitely many G-extensions of k. A natural question is then how large this specialization set is compared to the set of all G-extensions. We present evidence for the following conjecture: If f is not of a very special form, then the specialization set is "small" in a density sense (when counted by discriminant). For k=Q, we make concrete progress on this conjecture, essentially reducing it to the abc-conjecture. This connects to a related "sparsity" result in a recent joint work with Dèbes, Legrand and Neftin about the set of rational pullbacks of Galois covers, and also extends results by Granville about the special case of hyperelliptic curves.
As an application of our result, we show that in certain families of "twisted Galois covers", there are ``many" curves failing the Hasse principle.
This is joint work with F. Legrand.

Orateur: Joachim König (KAIST)

12:00 Lunch break

13:30 The Mumford--Tate conjecture implies the algebraic Sato--Tate conjecture

The famous Mumford—Tate conjecture asserts that, for every prime number $\ell$, Hodge cycles are $\mathbb{Q}_{\ell}$ linear combinations of Tate cycles, through Artin's caparisons theorems between Betti and étale cohomology. The algebraic Sato—Tate conjecture, introduced by Serre and developed by Banaszak and Kedlaya, is a powerful tool in order to prove new instances of the generalized Sato—Tate conjecture. This previous conjecture is related with the equidistribution of Frobenius traces.
Our main goal is to prove that the Mumford-Tate conjecture for an abelian variety $A$ implies the algebraic Sato-Tate conjecture for $A$. The relevance of this result lies mainly in the fact that the list of known cases of the Mumford-Tate conjecture was up to now a lot longer than the list of known cases of the algebraic Sato-Tate conjecture.
This is a joint work with Johan Commelin.

Orateur: Victoria Cantoral-Farfan (ICTP)

14:30 Coffee break

15:00 Counting points on algebraic varieties over the rational numbers

For both conceptual and practical reasons it is useful to have estimates on the number of points of algebraic varieties over Q, usually phrased in terms of asymptotics as the height of points increases. I will present a new such estimate, improving previous results by Bombieri, Pila, Heath-Brown, Browning, Salberger, Walsh and others. Time permitting, I will present an application to bounding the 2-torsion part of class groups of number fields.

This is joint work with Wouter Castryck, Raf Cluckers and Kien Huu Nguyen.

Orateur: Philip Dittmann (KU Leuven)