Commutative Algebra (Master in Mathematics, 2015)

Learning material (2015)

Lecture notes with exercise sheets from the lecture Commutative Algebra held in winter term 2015 in the Master in Mathematics at the University of Luxembourg.

On conjectures of Sato-Tate and Bruinier-Kohnen

in The Ramanujan Journal (2015), 36(3), 455-481

This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on ... [more ▼]

This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind-Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of Sato-Tate equidistribution for CM modular forms with an error term similar to that in the prime number theorem. [less ▲]

Automorphic Galois representations in the inverse Galois problem

Scientific Conference (2014, June 24)

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 ... [more ▼]

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. Many parts are joint work with Sara Arias-de-Reyna, Luis Dieulefait and Sug-Woo Shin. [less ▲]

Sur les représentations galoisiennes en théorie de Galois inverse

Presentation (2014, February 10)

On présentera un travail en commun avec Sara Arias-de-Reyna, Luis Dieulefait et Sug Woo Shin où nous réalisons, pour tout n pair et tout d, le groupe PGSp_n(F_{p^d}) ou PSp_n(F_{p^d}) comme groupe de ... [more ▼]

On présentera un travail en commun avec Sara Arias-de-Reyna, Luis Dieulefait et Sug Woo Shin où nous réalisons, pour tout n pair et tout d, le groupe PGSp_n(F_{p^d}) ou PSp_n(F_{p^d}) comme groupe de Galois sur les nombres rationnels, pour p dans un ensemble de densité positive. La démonstration est basée sur des systèmes compatibles de représentations galoisiennes ayant des propriétés locales spéciales. Au début de l'exposé on esquissera la stratégie; elle est basée sur un travail en commun avec Dieulefait dans le cas de la dimension 2. On expliquera ensuite l'existence d'un corps global et minimal tel que presque toute représentation résiduelle d'un système compatible peut être définie projectivement sur le corps résiduel. En plus, on énoncera une classification simple des représentations symplectiques contenant une transvection dans leur image. Finalement, on expliquera l'existence du système compatible désiré et comment utiliser des techniques de minoration du niveau pour obtenir notre application au problème de Galois inverse. [less ▲]

On Galois Representations of Weight One

in Documenta Mathematica (2014), 19

A two-dimensional Galois representation into the Hecke algebra of Katz modular forms of weight one over a finite field of characteristic p is constructed and is shown to be unramified at p in most cases.

Computations with Modular Forms: Proceedings of a Summer School and Conference, Heidelberg, August/September 2011

Book published by Springer (2014)

This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A ... [more ▼]

This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A key theme of the Conference and Summer School was the interplay between theory, algorithms and experiment. The 14 papers offer readers both, instructional courses on the latest algorithms for computing modular and automorphic forms, as well as original research articles reporting on the latest developments in the field. The three Summer School lectures provide an introduction to modern algorithms together with some theoretical background for computations of and with modular forms, including computing cohomology of arithmetic groups, algebraic automorphic forms, and overconvergent modular symbols. The 11 Conference papers cover a wide range of themes related to computations with modular forms, including lattice methods for algebraic modular forms on classical groups, a generalization of the Maeda conjecture, an efficient algorithm for special values of p-adic Rankin triple product L-functions, arithmetic aspects and experimental data of Bianchi groups, a theoretical study of the real Jacobian of modular curves, results on computing weight one modular forms, and more. [less ▲]

On symplectic Galois representations and the inverse Galois problem

Scientific Conference (2013, September 27)

Beschleunigung durch Abstraktion

Conference given outside the academic context (2013)

Symplectic Galois representations and applications to the inverse Galois problem

Presentation (2013, April 19)

Modular forms and the inverse Galois problem

Scientific Conference (2013, February 02)