SIMBa: Fundamental domains for the Bruhat-Tits tree for $\mathrm{GL}2\left(F{\mathfrak{p}}\right)$

Dimecres 8 de novembre se celebrarcelebrarà una xerrada del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Conferenciant: Eloi Torrents Juste.
Universitat: Universitat Autònoma de Barcelona.

Data: Dimecres, 8 de novembre de 2023.
Hora: 13:00, pausa pel cafè; 13:20, xerrada.
Lloc: UAB (aula petita CRM) i Zoom.
Idioma: Anglès.

Títol: Fundamental domains for the Bruhat-Tits tree for $\mathrm{GL}_2\left(F_{\mathfrak{p}}\right)$ .

Resum: The computation of fundamental domains of the Bruhat-Tits tree by the action of quaternionic groups allows the computation of harmonic cocycles on it. These are related to automorphic forms and from this fact are derived several applications, as for example the computation of points on Shimura curves and Heegner points on elliptic curves as done by M. Greenberg and later generalized by C. Franc and M. Masdeu.
In this talk we will review these concepts, and we will explain how to apply them in the computation of Heegner points on elliptic curves in cases where the Heegner hypothesis is not satisfied, and therefore the classical arquimedian construction of these points is difficult to compute.