Winfried Kohnen
(Universität Heidelberg)

23/06/2014, 10:00

Analytic-Additive Number Theory

We will give a survey on recent results about sign changes of Fourier coefficients of cusp forms in one and several variables.

Lilian Matthiesen
(Institut de Mathématiques de Jussieu)

23/06/2014, 11:30

Analytic-Additive Number Theory

The aim of this talk is to explain a strategy that allows us to bound the Fourier coefficients of a large class of not necessarily bounded multiplicative functions. The interest in this result lies in the fact that the strategy can be adapted to show that these multiplicative functions give rise to functions that are orthogonal to linear nilsequences when applying a `W-trick'. This, in turn,...

Catherine Goldstein
(Institut de mathématiques de Jussieu)

23/06/2014, 14:30

Analytic-Additive Number Theory

Contrarily to other parts of number theory, the history of analytic number theory often appears as a collection of particular, even isolated, episodes, focussing on Euler or Riemann or Hadamard and de La Vallée-Poussin. The talk will discuss some of these gems, as well as less well-known ones, and comment on the discontinuous character of their history.

Julio Andrade
(IHES)

23/06/2014, 15:45

Analytic-Additive Number Theory

In this talk I will explore some traditional problems of analytic number theory in the context of function fields over a finite field. Several such problems which are currently viewed as intractable can, in the function field scenario, be attacked with vastly different tools than those of traditional analytic number theory. The resulting theorems in the function field setting can be used to...

Alain Plagne
(École polytechnique)

23/06/2014, 17:15

Analytic-Additive Number Theory

Nous étudions des phénomènes de seuil en théorie additive des nombres. L'objet central est les pseudo puissances s-ièmes introduites par Erdos et Renyi en 1960. In 1975, Goguel a montré que, presque surement, une telle suite n'était pas une base asymptotique d'ordre s. On verra qu'elle est presque surement base d'ordre s+\epsilon. On étudie aussi la taille du plus petit complément additif de...

Gebhard Boeckle
(Heidelberg University)

24/06/2014, 09:00

Galois representations and modular forms

Recently Dorobisz, Eardley-Manoharmayum and Manoharmayum have proved abstract results (a) on the shape of possible deformation rings and (b) on the image of universal deformations of profinite groups, for representations into GL_n. The result regarding (a) were motivated by questions of Bleher, Chinburg and de Smit. We place these results in an axiomatic framework that in principle applies to...

François Brunault
(École normale supérieure de Lyon)

24/06/2014, 10:30

Galois representations and modular forms

An abelian variety defined over a number field is called strongly modular when its L-function is the product of L-functions of modular forms of weight 2. In this talk, we will show a weak version of Beilinson's conjectures for non-critical L-values of strongly modular abelian varieties. We will explain the interest of formulating an equivariant version of these conjectures (after Burns and...

Nicolas Billerey
(Université Clermont-Ferrand 2)

24/06/2014, 11:45

Galois representations and modular forms

In this talk, I'll give a modularity result for reducible mod l Galois representations. By analogy with the irreducible case, I'll state some questions regarding characterization and optimization of the different types of modular forms attached to such a given representation. Finally, I'll give an application of these results to the determination of an explicit lower bound for the highest...

Luis Dieulefait
(Universitat de Barcelona)

24/06/2014, 15:30

Galois representations and modular forms

This is joint work with Ariel Pacetti. We present generalizations to totally real number fields of the construction done by the speaker some years ago over Q that allows to connect to each other any given pair of newforms through chains of modular compatible systems of Galois representations. We also discuss applications of this, and we consider the case of abstract Galois representations and...

Gabor Wiese
(Université du Luxembourg)

24/06/2014, 17:00

Galois representations and modular forms

In the talk I will report on recent results on the inverse Galois problem based on compatible systems of Galois representations coming from modular and automorphic forms. The focus will be on ideas and strategies as well as the obstacles that are preventing us from proving much stronger theorems. In this context, the role of coefficient fields will be particularly highlighted. Most parts are...

Walter Ferrer-Santos
(Universidad de la Republica, Montevideo)

25/06/2014, 09:00

Noncommutative algebra

An involutory Hopf algebra is a Hopf algebra whose antipode squared equals the identity, $S^2=\operatorname{id}$.
The identity map is an automorphism of Hopf algebras, hence it is tempting to substitute $\operatorname{id} \mapsto \sigma$ where $\sigma$ is an arbitray Hopf morphism and consider Hopf algebras whose antipode (that is an antimorphism of Hopf algebras) squared is the square of...

Alberto Facchini
(University of Padova)

25/06/2014, 10:30

Noncommutative algebra

We will describe direct-sum decompositions and direct-product decompositions for some classes of modules. We will be mainly interested in direct sums and direct products of modules whose endomorphism rings have at most two maximal ideals.

Patrick Solé
(Telecom ParisTech)

25/06/2014, 11:45

Noncommutative algebra

In this article we introduce skew generalized quasi-cyclic codes over finite field $F$ with Galois automorphism $\theta$. This is a generalization of quasi-cyclic codes and skew polynomial codes.
These codes have an added advantage over quasi-cyclic codes, since the length of the code $C$ need not be a multiple of the index of $C$. After a brief description of the skew polynomial ring...

Olivier Wittenberg
(École normale supérieure)

26/06/2014, 09:00

Arithmetic geometry and Galois theory

Si X est une variété projective et lisse définie sur un corps de nombres, la ``méthode des fibrations'' pour étudier l'ensemble des points rationnels de X ou le groupe de Chow des zéro-cycles de X vise à ramener les questions que l'on pose pour X (par exemple: existence d'un point ou d'un zéro-cycle de degré 1) aux mêmes questions pour les fibres d'un morphisme dominant f:X->P^1. Le but de...

David Harbater
(University of Pennsylvania)

26/06/2014, 10:30

Arithmetic geometry and Galois theory

The Oort conjecture states that every cyclic branched cover of curves in characteristic p can be lifted to such a cover in characteristic zero. This raises the more general question of which finite groups G have the property that every G-Galois branched cover of curves in characteristic p can be lifted to characteristic zero. While this can be viewed as analogous to the inverse Galois...

David Harari
(Université Paris-Sud)

26/06/2014, 11:45

Let T be an algebraic torus defined over a number field K. In the case of a number field, obstructions to local-global principles for T are well understood thanks to work by Voskresenskii and Sansuc. We consider the case K=k(t) for different fields k (quasi-finite, p-adic) and extend the classical results in this context.

Lorenzo Ramero
(Université Lille 1)

26/06/2014, 15:30

Arithmetic geometry and Galois theory

Scholze's theory of perfectoid rings and perfectoid spaces is rather recent, but it has already had some spectacular applications to étale cohomology, p-adic Hosge theory and p-adic representations. I will present a generalization of this theory that I am developing in collaboration with Ofer Gabber. I will also explain the questions that have led us to this generalization.

Aaron Silberstein
(University of Pennsylvania)

26/06/2014, 17:00

Arithmetic geometry and Galois theory

Given a field $K$, finitely generated and of transcendence degree $2$ over the algebraic closure of a prime field, we may now reconstruct $K$ from the maximal $2$-step nilpotent pro-$\ell$ quotient of its absolute Galois group. This allows us to construct a complete (albeit countably infinite) set of geometric obstructions for an element of the Grothendieck-Teichmüller group to come from an...

Alexander Vishik
(University of Nottingham)

27/06/2014, 09:00

Quadratic forms

This is a joint work with Alexander Smirnov. I will describe a new homotopic approach to the classification of torsors of algebraic Groups. It extends the approach of Morel-Voevodsky, where torsors are interpreted as Hom’s to the classifying space of the group in the A^1-homotopy category of Morel-Voevodsky. In the case of the orthogonal group O(n), we introduce new invariants: “Subtle...

Bruno Kahn
(Institut de Mathématiques de Jussieu)

27/06/2014, 10:30

Quadratic forms

On définit une notion de réciprocité sur les préfaisceaux avec transferts (PST) de Voevodsky. Pour cela, on enrichit les groupes de 0-cycles avec module de Kerz-Saito en leur conférant une structure de PST. Les PST invariants par homotopie sont à réciprocité, ainsi que ceux représentables par un groupe algébrique commutatif : ce dernier point généralise un théorème classique de Rosenlicht qui...

Jean-Pierre Tignol
(Université catholique de Louvain)

27/06/2014, 11:45

Quadratic forms

La dimension, le discriminant, et l'invariant de Clifford sont des invariants classiques des formes quadratiques, qui s'étendent au contexte plus général des algèbres centrales simples à involution orthogonale. Sous certaines conditions, on peut aussi définir un invariant d'Arason; mais contrairement à ce qui se passe pour les formes quadratiques, celui-ci n'est pas toujours représenté par une...