An Application of Maeda's Conjecture to the Inverse Galois Problem

in Mathematical Research Letters (2013), 20(5), 985-993

It is shown that Maeda's conjecture on eigenforms of level 1 implies that for every positive even d and every p in a density-one set of primes, the simple group PSL_2(F_{p^d}) occurs as the Galois group ... [more ▼]

It is shown that Maeda's conjecture on eigenforms of level 1 implies that for every positive even d and every p in a density-one set of primes, the simple group PSL_2(F_{p^d}) occurs as the Galois group of a number field ramifying only at p. [less ▲]

Algèbre 1 (BASI filière mathématiques, 2013)

Learning material (2013)

Lecture notes written in French from the Algebra 1 lecture in the 1st term of the Bachelor programme BASI branch Mathematics at the University of Luxembourg. The lecture starts with preliminaries on logic ... [more ▼]

Lecture notes written in French from the Algebra 1 lecture in the 1st term of the Bachelor programme BASI branch Mathematics at the University of Luxembourg. The lecture starts with preliminaries on logic, sets and functions, it builds the natural numbers (almost) from the Peano axioms, then constructs the integers and the rationals. Groups and rings are introduced in that context. The most basic definitions and results from abstract linear algebra are also given. The course finishes with some basic group theory. [less ▲]

Winter School on Galois Theory, Volume 1

Book published by University of Luxembourg / Campus Kirchberg (2013)

Galois Theory plays a key role in many mathematical disciplines, such as number theory, algebra, topology, and geometry. This special volume of the Luxembourg based peer-reviewed mathematics journal ... [more ▼]

Galois Theory plays a key role in many mathematical disciplines, such as number theory, algebra, topology, and geometry. This special volume of the Luxembourg based peer-reviewed mathematics journal "Travaux mathématiques" unites two instructional texts that have grown out of lectures delivered at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. The contribution by Wulf-Dieter Geyer is about "Field Theory". It can be considered as a textbook in its own right. It manages to start at the level that any student possesses after any introductory algebra course and nevertheless to lead the reader to very advanced field theory at the frontier of current research, and to cover a wealth of material. Many examples are contained, which nicely enlighten the presented concepts, very often providing counterexamples that show why certain hypotheses are necessary. One also finds a chapter on the history of field theory as well as other historical remarks throughout the text. The second contribution addresses "Profinite Groups". It is written by Luis Ribes, who is the author of two standard books on this subject. Being necessarily much shorter than the two books, it has the feature of presenting an overview stressing the main concepts and the links with Galois Theory. Since for those proofs which are not included precise references are given, the notes, due to their conciseness and nevertheless great amount of material, constitute an excellent starting point for any Master or PhD student willing to learn this subject. [less ▲]

On modular Galois representations modulo prime powers

in International Journal of Number Theory (2013), 9(1), 91--113

We study modular Galois representations mod p^m. We show that there are three progressively weaker notions of modularity for a Galois representation mod p^m: we have named these `strongly', `weakly', and ... [more ▼]

We study modular Galois representations mod p^m. We show that there are three progressively weaker notions of modularity for a Galois representation mod p^m: we have named these `strongly', `weakly', and `dc-weakly' modular. Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M. Using results of Hida we display a `stripping-of-powers of p away from the level' type of result: A mod p^m strongly modular representation of some level Np^r is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod p^m corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod p^m to any `dc-weak' eigenform, and hence to any eigenform mod p^m in any of the three senses. We show that the three notions of modularity coincide when m=1 (as well as in other, particular cases), but not in general. [less ▲]

Symplectic Galois representations and applications to the inverse Galois problem

Presentation (2012, May 30)

Modulare Galois-Darstellungen und Computeralgebra

Scientific Conference (2012, May 12)

Symplectic Galois representations and applications to the inverse Galois problem

Presentation (2012, January 19)

Lectures on Modular Galois Representations Modulo Prime Powers

Presentation (2011, December)

A computational study of the asymptotic behaviour of coefficient fields of modular forms

in Actes de la Conférence ``Théorie des Nombres et Applications'' (2011)

The article motivates, presents and describes large computer calculations concerning the asymptotic behaviour of arithmetic properties of coefficient fields of modular forms. The observations suggest ... [more ▼]

The article motivates, presents and describes large computer calculations concerning the asymptotic behaviour of arithmetic properties of coefficient fields of modular forms. The observations suggest certain patterns, which deserve further study. [less ▲]

Computing congruences of modular forms and Galois representations modulo prime powers

in Arithmetic, geometry, cryptography and coding theory 2009 (2010)

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two ... [more ▼]

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented. [less ▲]