The Algebra and Number Theory group of the University of Luxembourg hosts three seminars. Some of them are currently being held in a hybrid format.

- The
**Luxembourg Number Theory Seminar**hosts invited speakers. - In the
**Work in Progress Seminar**members of the group present their work in progress; it alternates with the Number Theory Seminar. - In the
**Reading Seminar**we study a topic together; during term time the seminar takes place weekly.

Everyone is invited to attend! For more information, you can contact Andrea Conti or Bryan Advocaat.

During the Summer Semester, the seminar typically meets on __Wednesdays at 10:30__. Unless otherwise specified, talks take place in room **MNO 1.010**. You can find below a collection of abstracts.

Date (Room) | Speaker | Title |

27/5/2024, 10:30 | Sara Checcoli (Institut Fourier, Grenoble) | tba |

22/5/2024, 10:30 | Francesco Campagna (Clermont-Auvergne Université) | tba |

8/5/2024, 10:30, chalk room | Julian Quast (University of Duisburg-Essen) | On local Galois deformation rings |

Thursday, 25/4/2024, 10:45 | Leolin Nkuete (Hausdorff Center for Mathematics, Bonn) | Maximal Curves of Genus 5 over Finite Fields |

tba | Wushi Goldring (Stockholm University) | tba |

13/3/2024, 10:30 | Alireza Shavali (Heidelberg University) | On the endomorphism algebra of abelian varieties associated with Hilbert modular forms |

28/2/2024, 10:30 | Oussama Hamza (Western University, Canada) | On extensions of number fields with given quadratic algebras and cohomology |

21/2/2024, 10:30 | Antonella Perucca (uni.lu) | Probabilities for primitive roots |

7/2/2024, 10:30 | Alexandre Maksoud (Paderborn University) | Adjoint p-adic L-functions of cuspforms |

9/1/2024, 10:30, MNO 1.020 | Lea Terracini (Università di Torino) | Bogomolov property for some modular Galois representations |

20/12/2023, MNO 1.030 | Samuele Anni, Adel Betina, Wushi Goldring, Chun Yin Hui | Number Theory Day |

28/11/2023, 14:00, MNO 1.020 | Anna Medvedovsky (MPIM Bonn) | Method of deep trace congruences, with applications |

9/11/2023, 14:00 | Fabio La Rosa (uni.lu) | Translation functors and the trace formula II |

31/10/2023, 14:00, MNO 1.040 | Neelam Kandhil (MPIM Bonn) | On linear independence of Dirichlet L values |

25/10/2023, 14:00 | Fabio La Rosa (uni.lu) | Translation functors and the trace formula I |

4/10/2023, 14:00 | Gabor Wiese (uni.lu) | Entanglement of Modular Forms |

The seminar will take place on __Tuesdays from 10:30 to 12:00__ in Room __MNO 1.010__, unless otherwise specified. The program can be found here.

Date (Room) | Speaker | Title |

10/10/2023 | Andrea Conti | Overview |

17/10/2023 | Antigona Pajaziti | Definitions and first properties |

27/10/2023 | Mingkun Liu | Lie algebras |

7/11/2023 | Flavio Perissinotto | |

14/11/2023 | Bryan Advocaat | |

24/11/2023 | Clifford Chan | |

28/11/2023 | Nathaniel Sagman |

**Leolin Nkuete (Hausdorff Center for Mathematics, Bonn)** *Maximal Curves of Genus 5 over Finite Fields*

A curve of genus g over a finite field F_q is called a maximal curve if its number of F_q-rational points reaches the upper Hasse-Weil-Serre bound q+1+ g[2\sqrt{q}]. In this talk we will investigate the existence of a maximal curve of genus 5 over F_q.

**Julian Quast (University of Duisburg-Essen)** *On local Galois deformation rings*

In joint work with Vytautas Paskunas, we show that the universal framed deformation ring of an arbitrary mod p representation of the absolute Galois group of a p-adic local field valued in a possibly disconnected reductive group G is flat, local complete intersection and of the expected dimension. In particular, any such mod p representation has a lift to characteristic 0. The work extends results of Böckle, Iyengar and Paskunas in the case G=GL_n. We give an overview of the proof of this main result.

**Alireza Shavali (Heidelberg University)** *On the endomorphism algebra of abelian varieties associated with Hilbert modular forms*

For every modular newform, Ribet and Momose construct a division algebra whose group of units approximates the image of the l-adic Galois representation associated to the modular form, for all l. One can compute the class of this division algebra in the Brauer group by explicit formulas found by Quer for weight 2 and Ghate-Jimenez-Quer for general weight, in terms of the Fourier coefficients of the modular form. Nekovar generalizes the construction of Ribet and Momose to Hilbert modular forms. In this talk we will generalize Quer's formula to Hilbert modular forms (in the parallel weight two case) under the assumption that the base field is an odd degree extension of Q.

**Oussama Hamza (Western University, Canada)** *On extensions of number fields with given quadratic algebras and cohomology*

At the beginning of the century, Labute and Minac introduced a criterion, on presentations of pro-p groups, ensuring that the cohomological dimension is two. Groups with presentations satisfying this condition are called mild.

In this talk, we introduce a new criterion on the presentation of finitely presented pro-p groups which allows us to compute their cohomology groups and infer quotients of mild groups of cohomological dimension strictly larger than two.
We interpret these groups as Galois groups over p-rational fields with prescribed ramification and splitting.

**Antonella Perucca (uni.lu)** *Probabilities for primitive roots*

Primitive roots are a basic concept in modular arithmetic hence there are no specific prerequisites for this talk (the number theory facts can easily be taken as black boxes). We present the probability expectations on primitive roots that arise from Artin's conjecture and we address what we call the Artin paradox. We then discuss the algorithms that search for a primitive root by testing candidates.

**Alexandre Maksoud (University of Paderborn)** *Adjoint p-adic L-functions of cuspforms*

In this talk, we give an alternative construction to Bellaïche's adjoint p-adic L-function on the ordinary locus of the eigencurve, which we also relate to characteristic ideals of Greenberg-Selmer groups. If time permits, we also discuss the calculation of its special values at classical weight one points, and applications to the variation of Iwasawa invariants attached to p-adic Rankin-Selberg L-functions.

**Lea Terracini (Università di Torino)** *Bogomolov property for some modular Galois representations*

In 2013 P. Habegger proved the Bogomolov property for the field generated over $\mathbb{Q}$ by the torsion points of a rational elliptic curve. We explore the possibility of applying the same strategy of proof to the case of field extensions cut out by some modular Galois representations.

**Anna Medvedovsky (MPIM Bonn)** *Method of deep trace congruences, with applications*

I will discuss a method for counting mod-p eigensystems carried by subquotients of spaces of classical modular forms by establishing deeper p-power congruences between traces of prime-power Hecke operators. The motivating application (joint with Samuele Anni and Alexandru Ghitza) is to refining the dimension split between Atkin-Lehner plus-minus eigenspaces in level p to account for mod-p congruences. The method generalizes to give partial results (or, depending on perspective, evidence) towards, among other things, establishing higher congruences between p-new forms in the same weight, recently discovered by Conti and Gräf.

**Neelam Kandhil (MPIM Bonn)** *On linear independence of Dirichlet L values*

The study of linear independence of L(k,chi) for a fixed integer k>1 and varying chi depends critically on the parity of k vis-à-vis chi. Several authors have explored this phenomenon for Dirichlet characters chi with fixed modulus and having the same parity as k. We extend this investigation to families of Dirichlet characters modulo distinct pairwise co-prime natural numbers across arbitrary number fields. In the process, we determine the dimension of the multi-dimensional generalization of cotangent values and the sum of generalized Chowla-Milnor spaces over the linearly disjoint number fields.

**Fabio La Rosa (uni.lu)** *Translation functors and the trace formula I*

I will begin by recalling the main ideas involved in the proof of the trace formula for connected semisimple anisotropic algebraic groups defined over the rational numbers. The emphasis will be on the spectral side of the trace formula: the proof that the right regular representation of the adelic points of a group as above defines a trace-class operator and the ensuing decomposition of the right regular representations into automorphic representations. I will then explain the restricted tensor product decomposition of an automorphic representation into an Archimedean and a non-Archimedean part and complete the talk by explaining the main properties of the Archimedean objects involved in the decomposition.

**Fabio La Rosa (uni.lu)** *Translation functors and the trace formula II*

I will propose a way to combine the theory of translation functors with the trace formula to study automorphic representations of connected semisimple anisotropic algebraic groups over the rational numbers whose Archimedean component is a limit of discrete series. I will explain the main ideas of the derivation of a trace formula which, modulo a conjecture on the decomposition of the tensor product of a limit of discrete series with a finite-dimensional representation into basic representations, allows to isolate the non-Archimedean parts of a finite family of C-algebraic automorphic representations containing the ones whose Archimedean component is a given limit of discrete series.

**Gabor Wiese (uni.lu)** *Entanglement of Modular Forms*

In this talk, I will survey joint work in progress with Samuele Anni and Luis Dieulefait on entanglement for modular forms. We approach the question from a group theoretic point of view. The main objects of study are non-abelian entanglement fields: we show that any such has to arise from a weight one modular form in a very precise way.

Last modification: 2 October 2023.