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**the group discusses its work in progress; it alternates with the Number Theory Seminar. - In the
**Research Seminar**the group members study a topic together; during term time the seminar takes place weekly.

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

During the Winter Semester, this seminar typically meets on __Thursdays at 14:00__. Unless otherwise specified, talks take place in the "chalk room" on the first floor of MNO. You will find below a collection of abstracts.

Date (Room) | Speaker | Title |

13/10/2022, 14:00 | Antonella Perucca | Unified treatment of Artin-type problems |

20/10/2022, 14:00 | Bryan Advocaat | A conjecture of Coleman on the Eisenstein family |

27/10/2022, 14:00 | Emiliano Torti | On the existence of rigid analytic families of Galois-stable lattices in trianguline representations |

17/11/2022, 14:00 (6A) | Stevan Gajović | Rational points on curves and variations on the method of Chabauty and Coleman |

24/11/2022, 14:00 (MNO 1.020) | Riccardo Pengo | On the Northcott property for special values of L-functions |

16/12/2022, 14:00 (MNO 1.050) | Lassina Dembelé, Harald Helfgott, Igor Shparlinski, Lola Thompson | Number Theory Day |

13/4/2023, 13:00 (MNO 6A) | Gerard van der Geer | On isogenies of abelian varieties |

8/6/2022, 11:00 (MNO 6A) | Daniel Gil-Muñoz | Non-Galois Kummer extensions with Hopf-Galois theory |

The seminar will take place on __Thursdays from 11:30 to 13:00__. Unless otherwise specified, talks take place in the "chalk room" on the first floor of MNO. Here is the program of the seminar.

Date (Room) | Speaker | Title |

27/09/2022 | Andrea Conti | Overview I |

11/10/2022 | Gabor Wiese | Overview II |

18/10/2022 | Flavio Perissinotto | Group Cohomology I |

25/10/2022 | Fabio La Rosa | |

3/11/2022 | Bryan Advocaat | |

10/11/2022 | Clifford Chan | |

15/11/2022 | Clifford Chan, Gabor Wiese | |

22/11/2022 | Gabor Wiese, Pietro Sgobba | |

29/11/2022 | Pietro Sgobba | |

6/12/2022 | Bryan Advocaat | |

13/12/2022 | Bryan Advocaat |

**Daniel Gil-Muñoz** (Charles University in Prague) *Non-Galois Kummer extensions with Hopf-Galois theory*

Hopf-Galois theory is a generalization of Galois theory with the use of Hopf algebras. Galois extensions can be characterized uniquely in terms of the group algebra of the Galois group and its classical action on the top field. If in this equivalent definition we replace the Galois group algebra by an arbitrary Hopf algebra and the classical action by a linear action of the Hopf algebra, we obtain the notion of Hopf-Galois extension. In this talk we shall view a comprehensive introduction of Hopf-Galois theory and we will show how to use it to generalize Kummer theory for Galois extensions. Namely, we will see how to rewrite the Kummer condition for Galois extensions so as to define a notion of Kummer Hopf-Galois extension. This perspective can be used to characterize simple radical degree n extensions of a field K that are linearly disjoint with the extension of K generated by the n-th roots of the unity, and can be extended to a fairly general class of radical extensions of K.

**Antonella Perucca** (uni.lu) *Unified treatment of Artin-type problems*

Artin's Primitive Root Conjecture dates back to 1927. In its most basic form it states that for an integer $a$ (different from $0,1,-1$) that is not a square there exist infinitely many primes $p$ such that $(a \bmod p)$ generates the multiplicative group at $p$. In 1967, Hooley proved this conjecture relying on GRH, and since then many variants have been considered. In this talk we present a joint work with Järviniemi, where we explain how to deal with a question more general than Artin's Conjecture that unifies several previously considered variants. Thanks to this work, we are able to prove some related results (jointly with Sgobba), while dealing in full generality, namely working over any number field $K$ and with finitely many subgroups of $K^\times$ of positive rank.

**Emiliano Torti** (uni.lu) *On the existence of rigid analytic families of Galois-stable lattices in trianguline representations*

In this talk, we will define rigid analytic families of representations of topological compact groups and we will study the existence of integral subfamilies. The obtained results will be applied to the study of lattices in trianguline (in particular, semi-stable and crystalline) representations of generic dimension, which is one of the main topics in integral $p$-adic Hodge theory.

**Stevan Gajović** (Max Planck Institute for Mathematics, Bonn) *Rational points on curves and variations on the method of Chabauty and Coleman*

In this talk, we present various problems that amount to computing rational points on curves and briefly explain how, in certain cases, we determine rational points on curves using variations on the method of Chabauty and Coleman.

**Riccardo Pengo** (Max Planck Institute for Mathematics, Bonn) *On the Northcott property for special values of L-functions*

According to Northcott's theorem, each set of algebraic numbers whose height and degree are bounded is finite. Analogous finiteness properties are also satisfied by many other heights, as for instance the Faltings height. Given the many (expected and proven) links between heights and special values of L-functions (with the BSD conjecture as the most remarkable example), it is natural to ask whether the special values of an L-functions satisfy a Northcott property. In this talk, based on a joint work with Fabien Pazuki, and on another joint work in progress with Jerson Caro and Fabien Pazuki, we will show how this Northcott property is often satisfied at the left of the critical strip, and not satisfied on the right. We will also overview the links between these Northcott properties and those of the motivic heights defined by Kato, and also some effective aspects of our work, which aim at giving some explicit bounds for the cardinality of the finite sets that we come across.

Last modification: 8 June 2023.