SIMBa: An Illustrative Example of Class Field Theory: Abelian Extensions of $\mathbb{Q}(i)$ via the Lemniscatic curve

10/06/2024 by blocmat Deixa un comentari

Dimecres 12 de juny se celebrarà una xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Conferenciant: Josu Pérez Zarraonandía.
Universitat / Institut de recerca: Universitat de Barcelona.

Data: Dimecres, 12 de juny de 2024.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UB (FMI aula IA) i Zoom.
Idioma: Anglès.

Títol: An Illustrative Example of Class Field Theory: Abelian Extensions of $\mathbb{Q}(i)$ via the Lemniscatic curve

Resum: The class field theory of $\mathbb{Q}$ is elegantly described in terms of the cyclotomic extensions $\mathbb{Q}\left(\zeta_n\right)$ . The process of adjoining roots of unity to $\mathbb{Q}$ yields abelian extensions, with any abelian extension of $\mathbb{Q}$ contained in at least one such extension. In general, the main theorem of class field theory asserts that the abelian extensions of a number field are described by the closed subgroups of its idele class group with finite index.
While this formulation represents a significant milestone in 20thcentury number theory, the adelic approach to class field theory abstracts away from an explicit description of abelian extensions in terms of algebraic integers. The aim of this talk is to explain how we can recover this explicit description for the abelian extensions of $\mathbb{Q}(i)$ using torsion points on the lemniscatic curve.