## 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 17/04/2018, 11:30 Gerard van der Geer Modular forms and curves over finite fields 25/04/2018, 09:00 Marc Houben (Utrecht) Zeta functions of dynamically affine maps in positive characteristic 26/04/2018, 14:00 (MNO 1.050) Matthew Bisatt (King's College) Root numbers of abelian varieties 02/05/2018, 10:00 (MNO 6B) François Legrand (Dresden) On some variants of the (Regular) Inverse Galois Problem 02/05/2018, 14:00 (MNO 6B) Andrea Ferraguti (Cambridge) Permutation rational functions via Chebotarev density theorem 02/05/2018, 15:00 (MNO 6B) Francesco Campagna (Leiden) Cyclic Reduction of Elliptic Curves 22/08/2018, 15:00 (MNO 6B) Carl Wang-Erickson (Imperial) On a question of Coleman and Greenberg

#### 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 Level raising for deformation rings (part 1) 06/03/2018, 11:30 Shaunak Deo Level raising for deformation rings (part 2) 06/03/2018, 14:00 Antonella Perucca Artin's Conjecture and related problems 20/03/2018, 11:30 Gabor Wiese Remarks on newform theory 17/04/2018, 14:00 Daniel Berhanu On hypergeometric Bernoulli numbers 08/05/2018, 10:30 Luca Notarnicola Lattice reduction 15/05/2018, 11:30 Gergely Kiss Discrete Fuglede conjecture and Pompeiu problem 15/05/2018, 14:00 Daniel Berhanu Classical newform theory 19/06/2018, 14:00 (MSA 3.230) Alexander D. Rahm Bianchi orbifolds in explicit computations and applications

#### 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) 27/02/2018, 14:00 Shaunak Deo Hilbert modular forms after Diamond-Sasaki (part 2) 02/03/2018, 14:30 Mariagiulia De Maria Hilbert modular forms after Diamond-Sasaki (part 3) 09/03/2018, 14:00 Mariagiulia De Maria Hilbert modular forms after Diamond-Sasaki (part 4) 13/03/2018, 14:00 Daniel Berhanu Hilbert modular forms after Diamond-Sasaki (part 5) 24/04/2018, 11:30 Shaunak Deo Hilbert modular forms after Diamond-Sasaki (part 6) 24/04/2018, 14:00 Mariagiulia De Maria Hilbert modular forms after Diamond-Sasaki (part 7)

#### 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.

Marc Houben (Utrecht) Zeta functions of dynamically affine maps in positive characteristic

We study the dynamics of endomorphisms of algebraic varieties over an algebraically closed field of positive characteristic. Of particular interest are so-called dynamically affine maps: morphisms arising as a finite quotient of an endomorphism of an algebraic group. For dynamically affine maps on the projective line, we present a dichotomy rationality/non-holonomicity for the corresponding Artin-Mazur zeta function.

Matthew Bisatt (King's College) Root numbers of abelian varieties

Consider an elliptic curve over a number field. The set of rational points of is well known to have the structure of a finitely generated abelian group and its rank is famously predicted to equal the order of vanishing of its L-function. This conjecture however assumes that the L-function has analytic continuation which is not always known. To avoid this, we introduce the root number which is independent of the L-function and conjecturally controls whether the rank is odd or even. In particular, if this object tells us the rank is odd then this implies that the elliptic curve has infinitely many rational points.
I will discuss how one computes the root number in practice for elliptic curves and generalise this to their higher dimensional analogues: abelian varieties. As an application, I give an example of an abelian variety such that every quadratic twist (conjecturally) has positive rank.

François Legrand (Dresden) On some variants of the (Regular) Inverse Galois Problem

The Inverse Galois Problem (IGP) asks whether every finite group G occurs as the Galois group of a Galois extension of Q. The geometric approach, called the Regular Inverse Galois Problem (RIGP), consists in producing a regular Galois extension of Q(T) of group G, and then in specializing the indeterminate T properly (by using Hilbert's Irreducibility Theorem). Despite many efforts in the last decades, these two questions remain widely open.
In this talk, I shall discuss a strong version of the RIGP which consists in realizing every finite group G as the Galois group of a regular Galois extension of Q(T) which provides all the Galois extensions of Q of group G by specialization. Time permitting, I shall also discuss a weak version of the IGP and the RIGP where the notion of Galois group is replaced by that of automorphism group.

Andrea Ferraguti (Cambridge) Permutation rational functions via Chebotarev density theorem

Constructing permutation functions of finite fields is a task of great interest in coding theory and cryptography. Permutation polynomials over finite fields have been completely classified up to degree 6, with "ad hoc" methods for every degree. In this talk, we present a general approach for classifying permutation rational functions of any degree that exploits a refined version of Chebotarev density theorem for function fields due to Kosters. We will show how to use the method to completely classify permutation rational functions of degree 3. This is joint work with Giacomo Micheli.

Alexander D. Rahm Bianchi orbifolds in explicit computations and applications

Bianchi orbifolds are at the crossroads of hyperbolic geometry, modular forms, harmonic analysis, heat kernels, knot theory and mathematical string theory; and provide meaningful examples for all of these fields of theory. Hyperbolic 3-space has a straight link to special linear matrix groups, being isomorphic to the quotient of the Lie group SL_2(C) modulo its maximal compact subgroup SU_2. Therefore it admits a natural action via isometries by discrete subgroups of SL_2(C), and the Bianchi groups are prototypical for non-co-compact subgroups of finite co-volume. A Bianchi orbifold is the quotient of hyperbolic 3-space by a Bianchi group, and it can be computed explicitly. I will present software that is available for this purpose - a user-friendly surface for one of them is currently under development at University of Luxembourg. Along with some colourful pictures of Bianchi orbifolds, I will also show examples of spaces of Bianchi modular forms, as well as examples for the Baum-Connes conjecture which links equivariant K-homology with operator K-theory, which can be computed with the software.

Carl Wang-Erickson (Imperial) On a question of Coleman and Greenberg

We present some joint work with Francesc Castella related to the following question of Ralph Greenberg and Robert Coleman. Let f be a classical p-ordinary cuspidal eigenform of weight at least two. It is well-understood that the associated irreducible 2-dimensional p-adic representation of the absolute Galois group of Q becomes reducible after restriction to a decomposition group at p. Greenberg asked for a characterization of those f such that this reducible representation splits. The following characterization has been proposed, with supporting evidence provided by Ghate and Vatsal: such f have complex multiplication.

Last modification: 16 August 2018.