Number Theory Days in Lille

Liste des Contributions

50. A Minkowski-type result for linearly independent subsets of ideal lattices

Gergely Harcos (Alfréd Rényi Institute of Mathematics)

08/07/2019 09:45

Analytic-additive number theory

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.

47. On the distribution of the maximum of partial sums of exponential sums

Youness Lamzouri (Université de Lorraine / York University)

08/07/2019 11:00

Analytic-additive number theory

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...

40. The mean value of cubic L-functions over function fields

Matilde Lalín (Université de Montréal)

08/07/2019 14:00

Analytic-additive number theory

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...

51. Bootstrapping partition regularity of linear systems

Tom Sanders (University of Oxford)

08/07/2019 15:30

Analytic-additive number theory

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...

45. Applications of the orbit method to the analysis of automorphic forms

Paul Nelson (ETH Zurich)

09/07/2019 09:45

Galois representations and modular 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.

55. Birch and Swinnerton-Dyer Formula for modular forms of arbitrary weight in the cases of analytic ranks 0 and 1

Dimitar Jetchev (EPFL)

09/07/2019 11:00

Galois representations and modular forms

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...

56. On a conjecture of Gross and explicit formula for Gross-Stark units

Mahesh Kakde (King's College London)

09/07/2019 14:00

Galois representations and modular forms

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...

52. Overconvergent cohomology and automorphic p-adic L-functions

Daniel Barrera (Universidad de Santiago de Chile)

09/07/2019 15:30

Galois representations and modular forms

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...

48. Quadratic forms and diophantine sets

Karim Johannes Becher (Universiteit Antwerpen)

10/07/2019 09:45

Quadratic forms

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...

41. Quadratic and Differential Forms over fields of characteristic 2

Roberto Aravire (Universidad Arturo Prat)

10/07/2019 11:00

Quadratic forms

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...

42. Pretorsion theories in general categories

Alberto Facchini (Università di Padova)

10/07/2019 14:00

Noncommutative algebra

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...

43. Galois theory for cowreaths

Blas Torrecillas (Universidad de Almería)

10/07/2019 15:30

Noncommutative algebra

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.

54. Hopf Ore extensions and the antipode

Walter Ferrer (Universidad de la República)

10/07/2019 16:45

Noncommutative algebra

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).

46. Image of two-dimensional pseudorepresentations

Jaclyn Lang (Université Paris 13)

11/07/2019 09:45

Galois representations and modular forms

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...

49. Density of specialization sets and Hasse principle in families of twisted Galois covers

Joachim König (KAIST)

11/07/2019 11:00

Arithmetic geometry and Galois theory

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...

53. The Mumford--Tate conjecture implies the algebraic Sato--Tate conjecture

Victoria Cantoral-Farfan (ICTP)

11/07/2019 13:30

Arithmetic geometry and Galois theory

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...

44. Counting points on algebraic varieties over the rational numbers

Philip Dittmann (KU Leuven)

11/07/2019 15:00

Arithmetic geometry and Galois theory

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...