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.

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, 16:00 (online, MSA 3.220) | Paul Pollack (University of Georgia) | tba |

03/11/2020, 14:00 (online, MSA 3.220) | Nikita Karpenko (University of Alberta) | An ultimate proof of Hoffmann-Totaro's conjecture |

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

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, 14:00 (MSA 3.220) | Sebastiano Tronto | tba |

10/11/2020, 14:00 (MSA 3.220) | Arturo Jaramillo | tba |

17/11/2020, 14:00 (MSA 3.220) | Andrea Conti | tba |

24/11/2020, 14:00 (MSA 3.220) | Antonella Perucca | Kummer theory for 1-dimensional tori defined over number fields |

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 (MSA 3.220) | ? | Non-archimedean rings |

03/11/2020, 10:30 (MSA 3.190) | ? | f-adic rings and Tate rings |

10/11/2020, 10:30 (MSA 3.220) | ? | Adic spectra of affinoid rings, I |

17/11/2020, 10:30 (MSA 3.220) | ? | Adic spectra of affinoid rings, II |

24/11/2020, 10:30 (MSA 3.220) | ? | Adic spaces, I |

01/12/2020, 10:30 (MSA 3.220) | ? | Adic spaces, II |

08/12/2020, 10:30 (MSA 3.220) | ? | From rigid analytic spaces and formal schemes to adic spaces |

**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*** 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.

**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.

**Antonella Perucca (uni.lu)*** Kummer theory for 1-dimensional tori defined over number fields*

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, thus covering also the non-split case. In this talk I will explain how to compute the degree of division fields for non-split 1-dimensional tori defined over a number field. The strategy is reducing 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. If time permits we will also address similar questions for products of 1-dimensional tori.

Last modification: 14 October 2020.