## Academic Year 2017/2018 - Number Theory Seminars

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.

#### Luxembourg Number Theory Seminar

 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 05/02/2018, 14:00 Lassina Dembélé Abelian varieties with everywhere good reduction over certain real quadratic fields 15/02/2018, 16:00 Samuele Anni (Bonn) Congruence graphs

#### Work in Progress Seminar

 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) 27/02/2018, 11:30 Shaunak Deo tba (part 1) 06/03/2018, 11:30 Shaunak Deo tba (part 2) 06/03/2018, 14:00 Antonella Perucca tba

#### Winter Term 2017/18: Research Seminar: Journal paper club

 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) 15/02/2018, 11:00 Mariagiulia De Maria Hilbert modular forms after Diamond-Sasaki (part 1) 23/02/2018, 11:00 Shaunak Deo Hilbert modular forms after Diamond-Sasaki (part 2) 27/02/2018, 14:00 tba Hilbert modular forms after Diamond-Sasaki (part 3)

#### Collection of abstracts

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.

Lassina Dembélé Abelian varieties with everywhere good reduction over certain real quadratic fields

In this talk, we give a complete classification of all abelian varieties with everywhere good reduction over the real quadratic fields of discriminant 53, 61, 73 and 89. This extends previous results of Fontaine and Schoof.

Last modification: 15 February 2018.