The Algebra and Number Theory group of the University of Luxembourg hosts three seminars.

- The
**Luxembourg Number Theory Seminar**hosts invited speakers and takes place occasionally. - In the
**Research Seminar**the group members study a topic together; during term time the seminar takes place weekly. - In the
**Work in Progress Seminar**the group discusses its work in progress; during term time the seminar takes place weekly.

Unless announced otherwise, the seminars take place in the "work place" in the 6th floor of the Maison du Nombre in Esch-Belval.

Everyone is invited to attend! For more information, please contact Gabor Wiese.

Date | Speaker | Title |

16/11/2016, 16:00 | Anna Medvedovsky (Brown/MPIM Bonn) | An explicit universal Galois representation on the mod-3 Hecke algebra |

07/12/2016, 16:00 | Sebastian Schönnenbeck (Aachen) | Computing in unit groups of orders with Voronoi's algorithm |

24/04/2017, 15:45 | Chun Yin Hui (Amsterdam) | On the semisimplicity of geometric monodromy action in F_\ell coefficients |

18/07/2017, 14:00, S6B | Bui Anh Tuan | The Euclidean algorithm on the modular tree |

Date | Speaker | Title |

28/09/2016, 16:00 | Shaunak Deo | Mod-p Hecke algebras |

26/10/2016, 14:15 | Laia Amorós | Images of Galois representations in mod p Hecke algebras |

02/11/2016, 14:15 | Shaunak Deo | The geometry of eigenvarieties at classical points of weight one: Part 1 |

09/11/2016, 16:00 | Shaunak Deo | The geometry of eigenvarieties at classical points of weight one: Part 2 |

14/12/2016, 16:00 | Alexander D. Rahm | Serre's solution to the congruence subgroup problem |

Our main reference for the beginning are Gouvea's Park City Lectures on *Deformations of Galois Representations*.

Week | Date | Speaker | Title |

1 | 06/03/2017, 14:00 | Shaunak Deo | Lectures 1-2 |

2 | 27/03/2017, 14:00 | Alexander D. Rahm | Lecture 3 |

3 | 03/04/2017, 14:00 | Mariagiulia De Maria | Lecture 4 |

4 | 10/04/2017, 10:00 | Gabor Wiese | Lecture 5 |

5 | 24/04/2017, 14:00 | Emiliano Torti | Lecture 6 |

6 | 08/05/2017, 14:00 | Jasper Van Hirtum | Lecture 7 |

7 | 15/05/2017, 14:00 | Shaunak Deo | Lecture 8 |

8 | 30/05/2017, 11:00 | Mariagiulia De Maria | Pseudo characters |

Reference for Weeks 1 to 9 is Jean-Pierre Serre : "Trees"/"Arbres, amalgames et SL_2".

Reference for Week 10 is a paper by Bellaiche and Chenevier

Reference for Week 11 is Laia's and Piermarco's joint preprint

Week | Date | Speaker | Title |

1 | 28/09/2016, 14:00 | Alexander D. Rahm | Introduction, Amalgams, Trees |

2 | 05/10/2016, 14:00 | Laia Amorós | Chapter 3 of J.-P. Serre's book | Trees and free groups |

3 | 12/10/2016, 14:15 | Shaunak Deo | Chapter 4 of J.-P. Serre's book | Trees and amalgams |

4 | 19/10/2016, 14:00 | Mariagiulia De Maria | Chapter 5 of J.-P. Serre's book | Structure of a group acting on a tree |

5 | 09/11/2016, 14:00 | Gabor Wiese | Chapter 6 of J.-P. Serre's book | Amalgams and fixed points |

6 | 16/11/2016, 14:00 | Shaunak Deo | Chapter II.1.1 - II.1.4 of J.-P. Serre's book | The tree of SL_2 over a local field |

7 | 23/11/2016, 14:00 | Gergely Kiss | Chapter II.1.5 - II.1.7 of J.-P. Serre's book | Ihara's and Nagao's theorems, connections with Tits systems |

8 | 30/11/2016, 14:00 | Mariagiulia De Maria | Chapter II.2.1 - II.2.5 of J.-P. Serre's book | Arithmetic subgroups of the groups GL_2 and SL_2 over a function field of one variable |

9 | 07/12/2016, 14:00 | Alexander D. Rahm | Trees, Amalgams and the Quillen conjecture |

10 | 14/12/2016, 14:15 | Shaunak Deo | Bellaiche's generalization of Ribet's Lemma using Bruhat-Tits trees |

11 | 21/12/2016, 14:15 | Laia Amorós | Applications of Bruhat-Tits trees to Shimura curves |

**Bui Anh Tuan** *The Euclidean algorithm on the modular tree *

The speaker will describe a Euclidean algorithm for elements of SL_2(Z), which is compatible with the action of SL_2(Z) on the modular tree, and this way equips the modular tree with an algorithm that encodes a contracting homotopy. This contracting homotopy can be used to control on the machine the amalgamated product SL_2(Z) *_{Gamma_0(p)} SL_2(Z) = SL_2(Z[1/p]) for a prime number p; and by the iterations described in Serre's book "Trees", the amalgamated product decomposition of SL_2(Z[1/m]) with m a composite integer number. This has yielded significant computational progress on the cohomology of SL_2(Z[1/m]).

**Anna Medvedovsky (Brown/MPIM Bonn)** *An explicit universal Galois representation on the mod-3 Hecke algebra *

We will construct a Galois representation attached to modular forms of level and all weights modulo 3 whose trace is universal. We will analyze its properties in detail and write down the representation explicitly in terms of matrices. Depending on time and interest we will discuss applications to distribution of Fourier coefficients of forms mod 3 and/or generalizations to p > 3, especially in the reducible level-one case. This work owes a great deal to published and unpublished work of Bellaiche and Serre on the case p = 2.

**Sebastian Schönnenbeck (Aachen)** *Computing in unit groups of orders with Voronoi's algorithm *

Unit groups of maximal orders in simple rational algebras (e.g. GL_n(Z) as the unit group of Mat_n(Z)) form an interesting class of (finitely presented) infinite groups. In general there are only few algorithms known for dealing with infinite groups; however, in this case we will see how one can employ a generalization of Voronoi's perfect form theory (which originally classified densest sphere packings) to answer of couple of basic questions about these groups. In particular we will compute a presentation, study how one can perform constructive membership and, if time permits, will construct a free resolution and classify the finite subgroups of such a unit group.

Last modification: 11 July 2017.