MATHEMATICAL CAREERS In Luxembourg and beyond

Diverse speeches and writings (2019)

Topics on modular Galois representations modulo prime powers

in Böckle, Gebhard; Decker, Wolfram; Malle, Gunter (Eds.) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory (2018)

This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes ... [more ▼]

This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences. [less ▲]

On the arithmetic of modular forms

Scientific Conference (2017, June 15)

In this short overview talk, we will stress the arithmetic significance of the coefficients of modular forms. This naturally leads to questions on the distribution of the coefficients in various senses ... [more ▼]

In this short overview talk, we will stress the arithmetic significance of the coefficients of modular forms. This naturally leads to questions on the distribution of the coefficients in various senses. We will briefly touch on some of them and state some open questions. The arithmetic information in the coefficients of a Hecke eigenform is summarised in the attached Galois representation. When studying its ramification properties, one notices that forms of weight one play a special role that we will explain. [less ▲]

On certain finiteness questions in the arithmetic of Galois representations

Presentation (2017, June 07)

Let p be a fixed prime number. It has been known for a long time that there are only finitely many Galois extensions K/Q with Galois group a finite irreducible subgroup of GL_2(F_p^bar) that are imaginary ... [more ▼]

Let p be a fixed prime number. It has been known for a long time that there are only finitely many Galois extensions K/Q with Galois group a finite irreducible subgroup of GL_2(F_p^bar) that are imaginary and unramified outside p. In contrast, there are infinitely many such with Galois group inside GL_2(Z_p^bar), even if one restricts to ones coming from modular forms (this restriction is believed to be local at p). It is tempting to ask what happens "in between" F_p^bar and Z_p^bar, i.e. whether there is still finiteness modulo fixed prime powers. In the talk, I will motivate and explain a conjecture made with Ian Kiming and Nadim Rustom stating that the set of such Galois extensions `modulo p^m' (a proper definition will be given in the talk) coming from modular forms is finite. I will present partial results and a relation of the finiteness conjecture to a strong question by Kevin Buzzard. The talk is based on joint work with Ian Kiming and Nadim Rustom. [less ▲]

Sur les représentations galoisiennes des formes modulaires de Hilbert de poids un en caractéristique p

Presentation (2017, January 19)

Dans cet exposé, je vais donner les idées principales de mon travail avec Mladen Dimitrov dans lequel nous démontrons que la représentation galoisienne associée à toute forme modulaire de Hilbert de poids ... [more ▼]

Dans cet exposé, je vais donner les idées principales de mon travail avec Mladen Dimitrov dans lequel nous démontrons que la représentation galoisienne associée à toute forme modulaire de Hilbert de poids un en caractéristique p et de niveau premier à p est non-ramifiée en p. Ce résultat s'applique notamment à des formes qui ne se relèvent pas en caractéristique zéro, et peut être vu comme la précision d'un aspect du poids dans la généralisation de la conjecture de modularité de Serre aux formes de Hilbert. [less ▲]

Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations

in Transactions of the American Mathematical Society (2017), 369

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine ... [more ▼]

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem. [less ▲]

On Galois representations of mod p Hilbert eigenforms of weight one

Scientific Conference (2016, August 16)

Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image

in Pacific Journal of Mathematics (2016), 281(1), 1-16

This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part is concerned with ... [more ▼]

This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part is concerned with symplectic Galois representations having a huge residual image, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. We prove a classification result on those subgroups of a general symplectic group over a finite field that contain a nontrivial transvection. Translating this group theoretic result into the language of symplectic representations whose image contains a nontrivial transvection, these fall into three very simply describable classes: the reducible ones, the induced ones and those with huge image. Using the idea of an (n,p)-group of Khare, Larsen and Savin we give simple conditions under which a symplectic Galois representation with coefficients in a finite field has a huge image. Finally, we combine this classification result with the main result of the first part to obtain a strenghtened application to the inverse Galois problem. [less ▲]

On certain finiteness questions in the arithmetic of modular forms

in Journal of the London Mathematical Society (2016), 94(2), 479-502

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness ... [more ▼]

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that, for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on \Gamma_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence. [less ▲]

Hilbertian fields and Galois representations

in Journal für die Reine und Angewandte Mathematik (2016), 712

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising ... [more ▼]

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In particular we settle a conjecture of Jarden on abelian varieties. [less ▲]