Seminarios T. de Números 

STATISTICS OF (ARAKELOV RAY) CLASS GROUPS AND ARITHMETICAL APPLICATIONS

SPEAKER: Carlo Pagano (Max Planck Institute for Mathematics, Bonn)

DATE & TIME: Tuesday, 19th November 2019 - 11:00

VENUE: Aula 420, Departamento de Matemáticas, UAM

ORGANISER: UAM - ICMAT

ABSTRACT: I will survey some recent work triggered by the dramatic
breakthrough of Alexander Smith on the Cohen--Lenstra and Goldfled's
conjecture. I will give an overview of some new ideas and notions coming
from Smith's work. From there I will explain a joint work with Peter
Koymans where we use some of these innovations to generalize a classical
result of Gauss on the 2-torsion of class groups from quadratic to
multi-quadratic fields; a joint work with Chan, Koymans and Milovic on the
Negative Pell equation; further progress on this and other Diophantine
problems obtained jointly with Peter Koymans. I will finally explain how a
non-archimedean version of Negative Pell leads naturally to wonder how
to extend the Cohen--Lenstra heuristics from class groups to ray class
groups. I will discuss the work I did on this topic for imaginary quadratic
fields, jointly with Sofos, and present some ongoing work with Alex Bartel
for more general families of number fields, where a notion of Arakelov ray
class group plays a crucial role.