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 |

19/10/2017, 11:30 | Nghia Thi Hieu Tran | Hochschild Cohomology of Algebras |

14/11/2017, 11:30 | Laia Amoros | Mumford curves covering Shimura curves and their fundamental domains |

05/12/2017, 11:30 | Sara Arias-de-Reyna | Ordinary abelian varieties and the Inverse Galois Problem |

Date | Speaker | Title |

26/09/2017, 14:00 | Gabor Wiese | Dihedral Galois deformations (part 1) |

03/10/2017, 11:30 | Shaunak Deo | Dihedral Galois deformations (part 2) |

10/10/2017, 14:00 | Emiliano Torti | On the congruences between cusp forms of weight k>=2 coming from level raising |

24/10/2017, 14:30 | no talk due to PhD defence of Gilles Becker | |

31/10/2017, 11:30 | Mariagiulia De Maria | Modular forms of weight one and Galois representations modulo prime powers |

07/11/2017, 11:30 | Gabor Wiese | Questions on modular forms modulo prime powers (1) |

14/11/2017, 14:00 | Alexander D. Rahm | Explicit non-trivial elements in algebraic K-theory |

28/11/2017, 11:30 | Gabor Wiese | Questions on modular forms modulo prime powers (2) |

Date | Speaker | Title |

26/09/2017, 11:30 | Antonella Perucca | Artin's Conjecture and related problems |

03/10/2017, 14:00 | Emiliano Torti | Congruences of modular forms (1) |

10/10/2017, 11:30 | Luca Notarnicola | Hidden subset sum problem |

17/10/2017, 14:00 | Alexander D. Rahm | Congruences of modular forms (2) |

24/10/2017, 11:30 | Mariagiulia De Maria | Congruences of modular forms (3) |

07/11/2017, 14:00 | Shaunak Deo | Congruences of modular forms (4) |

21/11/2017, 14:00 | Gabor Wiese | Congruences of modular forms (5) |

28/11/2017, 14:00 | Shaunak Deo | Determinants |

12/12/2017, 11:30 | Gabor Wiese | Unramified determinants |

12/12/2017, 14:00 | Mariagiulia De Maria | Determinants of weight one (after Calegari, Specter) |

**Sara Arias-de-Reyna (Sevilla)** * Ordinary abelian varieties and the Inverse Galois Problem *

Given an n-dimensional abelian variety A/Q which is principally polarised, we consider for each prime number \ell the representation of the absolute Galois group of the rational numbers, \rho_{A, \ell}: G_Q --> GSp_2n(F_\ell) attached to the \ell-torsion points of A. Provided the representation is surjective, we obtain a realisation of GSp_2n(F_\ell) as the Galois group of the finite extension Q(A[\ell])/Q, and the ramification type of a prime p in this extension can be read off from the type of reduction of A at p.

In this talk we address the question of producing tame Galois realisations of GSp_2n(F_\ell) by making use of those representations, and determine a series of local conditions ensuring tameness and surjectivity. In particular, we will work with abelian varieties ordinary at \ell. However, it is not clear how to set up the local conditions to force the existence of a global abelian variety (defined over Q) satisfying all of them simultaneously. In the cases when n \leq 3, we can make use of Jacobians of curves in a family, and deform the curves p-adically modulo a finite set of primes p to guarantee the local conditions hold, thus obtaining tame Galois realisations of GL_2(F_\ell)$, GSp_4(F_\ell)$ and GSp_6(F_\ell). For higher values of n, new ideas are required.

Last modification: 7 December 2017.