## Academic Year 2022/2023 - Number Theory Seminars

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.

#### Luxembourg Number Theory/Work-in-Progress Seminar

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 8/12/2022, 14:00 Flavio Perissinotto TBA 16/12/2022, 14:00 (MNO 1.050) Lassina Dembelé, Harald Helfgott, Igor Shparlinski, Lola Thompson Number Theory Day

#### Working Group: Class field theory and the Langlands correspondence for GL_1 (Winter semester 2022)

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

#### Collection of abstracts

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: 17 October 2022.