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; the seminar will partly be held online. - 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.

Everyone is invited to attend! For more information, please contact Alexandre Maksoud, Andrea Conti or Gabor Wiese.

This seminar typically meets on __Tuesdays at 14:00__. You will find below a collection of abstracts.

Date (Room) | Speaker | Title |

13/10/2020, 14:00 (MSA 3.220) | Gautier Ponsinet (Max Planck Institute for Mathematics) | Universal norms of p-adic Galois representations and the Fargues-Fontaine curve |

27/10/2020, 15:00 (online) | Paul Pollack (University of Georgia) | Thoughts on the order of a mod p |

03/11/2020, 14:00 (online) | Nikita Karpenko (University of Alberta) | An ultimate proof of Hoffmann-Totaro's conjecture |

01/12/2020, 14:00 (online) | Ariel Weiss (Einstein Institute of Mathematics) | Lafforgue pseudocharacters and the construction of Galois representations |

14/12/2020, all day (online) | Luxembourg Number Theory Day 2020 | |

04/02/2021, 14:00 (online) | Carlo Pagano (Max Planck Institute for Mathematics) | Distribution of fundamental units of varying real quadratic fields at a fixed place and Arakelov ray class groups |

This seminar alternates with the Luxembourg Number Theory Seminar. Each session lasts 60 to 90 minutes. You will find below a collection of abstracts.

Date (Room) | Speaker | Title |

29/09/2020, 14:00 (MSA 3.100) | Alexandre Maksoud | On Iwasawa theory and Bloch-Kato conjecture for Artin motives |

06/10/2020, 13:30 (MSA 3.220) | Daniel Berhanu Mamo | Eisenstein series and newform theory |

20/10/2020, 14:00 (MSA 3.220) | Arturo Jaramillo/Xiaochuan Yang | Selberg's theorem via Stein's method |

27/10/2020, 13:45 (online) | Sebastiano Tronto | Kummer theory for algebraic groups |

10/11/2020, 14:00 (online) | Arturo Jaramillo | A new probabilistic approach to Erdos-Kac theorem |

17/11/2020, 14:00 (online) | Andrea Conti | Trianguline representations and functoriality |

24/11/2020, 14:00 (online) | Flavio Perissinotto | Kummer theory for products of 1-dimensional tori |

08/12/2020, 14:00 (online) | Pietro Sgobba | On the order of the reductions of algebraic numbers |

23/02/2021, 14:00 (MNO 1.010) | Alexandre Maksoud | Generalized p-adic Stark conjectures and Iwasawa main conjectures for Artin motives |

02/03/2021, 14:00 (MNO 1.010) | Alexandre Maksoud | On the Iwasawa-theoretic conjectures for Rubin-Stark elements |

The seminar will take place on __Tuesdays from 10:30 to 12:30__ in room __MNO 1.010__. You can find here a tentative schedule.

Date (Room) | Speaker | Title |

16/02/2021, 10:30 (MNO 1.010) | Andrea Conti | Overview |

23/02/2021, 10:30 (MNO 1.010) | Bryan Advocaat | Hida theory |

You can find here a tentative schedule.

Date (Room) | Speaker | Title |

29/09/2020, 10:30 (MSA 3.100) | Andrea Conti | Introduction |

06/10/2020, 10:30 (MSA 3.220) | Bryan Advocaat | Valuations |

13/10/2020, 10:30 (MSA 3.220) | Alexandre Maksoud | Spectral and sober spaces |

20/10/2020, 10:30 (MSA 3.220) | Daniel Berhanu Mamo | Valuation spectra |

27/10/2020, 10:30 (online) | Alexandre Maksoud | Non-archimedean rings |

03/11/2020, 10:30 (online) | Emiliano Torti | f-adic rings and Tate rings |

10/11/2020, 10:30 (online) | Andrea Conti | Adic spectra of affinoid rings, I |

17/11/2020, 10:30 (online) | Bryan Advocaat | Adic spectra of affinoid rings, II |

24/11/2020, 10:30 (online) | Alexandre Maksoud | Adic spaces, I |

01/12/2020, 10:30 (online) | Andrea Conti | Adic spaces, II |

**Alexandre Maksoud (uni.lu)*** On Iwasawa theory and Bloch-Kato conjecture for Artin motives *

Inspired by the works of Perrin-Riou and of Benois we formulate a new cyclotomic Iwasawa Main Conjecture (IMC) for Artin motives, as well as an Exceptional Zeros Conjecture in this context. When the Artin representation is monomial, we show that our conjectures follow from the higher rank cyclotomic IMC recently introduced by Burns, Kurihara and Sano, together with the Iwasawa-theoretic Mazur-Rubin-Sano conjecture. We highlight some potential applications to a better understanding of special values of Artin L-functions at $s=0$, and to a conjecture on iterated p-adic integrals of Darmon-Lauder-Rotger.

**Daniel Berhanu Mamo (uni.lu)*** Eisenstein series and newform theory *

We set up a variant of strong multiplicity one theorems for Katz modular forms which admit a reducible mod $p$ Galois representation. An example that illustrates the main theorem will be presented.

**Gautier Ponsinet (Max Planck Institute for Mathematics)*** Universal norms of $p$-adic Galois representations and the Fargues-Fontaine curve*

In 1996, Coates and Greenberg computed explicitly the module of universal norms for abelian varieties in perfectoid field extensions. The computation of this module is essential to Iwasawa theory, notably to prove "control theorems" for Selmer groups generalising Mazur's foundational work on the Iwasawa theory of abelian varieties over $\mathbb{Z}_p$-extensions. Coates and Greenberg then raised the natural question on possible generalisations of their result to general motives. In this talk, I will present a new approach to this question relying on the classification of vector bundles over the Fargues-Fontaine curve, which enables to answer Coates and Greenberg's question affirmatively in new cases.

**Arturo Jaramillo and Xiaochuan Yang (uni.lu)*** Selberg's theorem via Stein's method *

Click here for the abstract.

**Sebastiano Tronto (uni.lu)*** Kummer theory for algebraic groups*

If G is a commutative and connected algebraic group over a number field K and A is a finitely generated and torsion-free subgroup of G(K), we can consider the set n^{-1}A={P in G(Kbar) | nP in A} of n-division points of A in an algebraic closure Kbar of K, for any positive integer n. The field extension K(n^{-1}A) of K generated by these points is Galois and it contains the n-torsion field K(G[n]) of G; we are interested in studying, among other things, its degree over this torsion field. This kind of extensions have been studied for example by Ribet [1], but the methods found in the literature are usually non-effective. In this talk I will present recent effective results of myself and Lombardo in the case of non-CM elliptic curves [2]. I will also outline a general framework, developped in [3], to reduce the study of these extensions to that of some arithmetic properties of A and certain properties of the Galois representations attached to G.

[1] Kennet A. Ribet, Kummer theory on extensions of abelian varieties by tori, Duke Mathematical Journal, 1979.

[2] Davide Lombardo and S. T., Explicit Kummer theory for elliptic curves, ArXiv preprint, 2019.

[3] S. T., Radical entanglement for elliptic curves, ArXiv preprint, 2020.

**Paul Pollack (University of Georgia)*** Thoughts on the order of a mod p*

I will discuss some old problems concerning the distribution of the orders of a modulo p, for a fixed integer a and varying primes p. The emphasis will be on connections between these problems and recent work of the speaker with K. Agrawal (UGA) --- on finite sets of integers at least one of which usually has a large order mod p --- and Z. Engberg (Wasatch Academy, UT) --- on arithmetic properties of Mersenne numbers.

**Nikita Karpenko (University of Alberta)*** An ultimate proof of Hoffmann-Totaro's conjecture*

We prove the last open case of the conjecture on the possible values of the first isotropy index of an anisotropic quadratic form over a field. It was initially stated by Detlev Hoffmann for fields of characteristic not 2 and then extended to arbitrary characteristic by Burt Totaro. The initial statement was proven by the speaker in 2002. In characteristic 2, the case of a totally singular quadratic form was done by Stephen Scully in 2015 and the nonsingular case by Eric Primozic in early 2019.

**Andrea Conti (uni.lu)*** Trianguline representations and functoriality*

I will introduce the notion of triangulinity for a p-adic representation of the absolute Galois group of a p-adic field, explaining how having this property at p is linked to modularity for a global Galois representation. As an application of this theory, I will present some known results and work in progress on how to prove that an n-dimensional global Galois representation is modular if and only if it is modular after composition with an algebraic representation of GL_n of dimension at least 2.

**Flavio Perissinotto (uni.lu)*** Kummer theory for products of 1-dimensional tori*

Classical Kummer theory concerns the study of cyclotomic-Kummer extensions, which can be interpreted as torsion and division fields for the multiplicative group. It is natural to address the same problems for general 1-dimensional tori and for products of 1-dimensional tori. In this talk I will explain how to compute the degree of division fields for products of 1-dimensional tori defined over a number field. The strategy is reducing first to a single 1-dimensional torus and then to the case of the multiplicative group, which amounts to understanding whether the splitting field of the torus is contained or not in the given division field.

**Ariel Weiss (Einstein Institute of Mathematics)*** Lafforgue pseudocharacters and the construction of Galois representations*

A key goal of the Langlands program is to attach Galois representations to automorphic representations. In general, there are two methods to construct these representations. The first, and the most effective, is to extract the Galois representation from the étale cohomology of a suitable Shimura variety. However, most Galois representations cannot be constructed in this way. The second, more general method is to construct the Galois representation, via its corresponding pseudocharacter, as a p-adic limit of Galois representations constructed using the first method.

In this talk, I will demonstrate how the second construction can be refined by using V. Lafforgue's G-pseudocharacters in place of classical pseudocharacters. As an application, I will prove that the Galois representations attached to certain irregular automorphic representations of U(a,b) are odd, generalising a result of Bellaïche-Chenevier in the regular case. This work is joint with Tobias Berger.

**Pietro Sgobba (uni.lu)*** On the order of the reductions of algebraic numbers*

Let K be a number field, and let G be a torsion-free and finitely generated subgroup of K*. We consider the order of the reduction (G mod p) for all but finitely many primes p of K. Assuming that K is normal over Q and without relying on GRH, we prove an asymptotic formula for the number of primes p for which the order of (G mod p) is divisible by a fixed integer. This result is a generalization of Pappalardi's work concerning the case K=Q.

**Carlo Pagano (Max Planck Institute for Mathematics)*** Distribution of fundamental units of varying real quadratic fields at a fixed place and Arakelov ray class groups*

The motivating question of this talk is the behavior of the fundamental units of varying real quadratic fields when reduced modulo a fixed integer. I will explain how this question is linked to the formulation of a Cohen-Lenstra heuristic for ray class groups of real quadratic fields. To this aim I will discuss first the case of imaginary quadratic fields and then, by introducing the notion of Arakelov ray class groups, the case of real quadratic fields. Along the way I will explain the new developments on Cohen-Lenstra at p=2, and some first results on the motivating question. The works discussed are joint with Alex Bartel, Peter Koymans and Efthymios Sofos.

**Alexandre Maksoud (uni.lu)*** Generalized p-adic Stark conjectures and Iwasawa main conjectures for Artin motives *

We will state a precise conjecture describing a generator of the Selmer group attached to Artin motives that are unramified at an odd prime p. This conjecture generalizes various classical conjectures in Iwasawa theory. We will then see how Coleman's theory applies to the study of Selmer groups. If time permits we will discuss potential applications to the Tamagawa Number Conjecture (aka. the Bloch-Kato Conjecture).

**Alexandre Maksoud (uni.lu)*** On the Iwasawa-theoretic conjectures for Rubin-Stark elements *

Burns, Kurihara and Sano recently proposed a strategy to tackle the equivariant Tamagawa Number Conjecture via the proof of Iwasawa-theoretic conjectures on Rubin-Stark elements. The aim of this talk is to introduce their work and to show the equivalence between their conjectures and our conjecture in the case where the Artin motive is induced from a character.

Last modification: 22 February 2021.