SEMINARIO TEORÍA DE NÚMEROS

Plectic insights on the BSD conjecture for higher rank elliptic curves

Ponente: Michele Fornea (CRM Barcelona)

Fecha: Lunes, 11 de marzo de 2024 - 12:30

Lugar: Aula 420, Módulo 17, Departamento de Matemáticas, UAM

Resumen:

Heegner points play a pivotal role in our understanding of the arithmetic of modular elliptic curves. They arise from CM points on Shimura curves and control the Mordell-Weil groups of elliptic curves of rank 1. The work of Bertolini, Darmon and their school has shown that p-adic methods can be successfully employed to generalize the definition of Heegner points to quadratic extensions that are not necessarily CM. Notably, Guitart, Masdeu, Sengun have defined and numerically computed Stark-Heegner points in great generality. Their computations strongly support the belief that Stark-Heegner points completely control the Mordell-Weil groups of elliptic curves of rank 1.

In this talk we will survey plectic generalizations of Stark-Heegner points developed in a series of articles with Darmon, Gehrmann, Guitart and Masdeu. These plectic Stark-Heegner points were inspired by Nekovar-Scholl’s plectic conjectures and should help illuminate the arithmetic intricacies of higher rank elliptic curves. Their construction is p-adic, cohomological, and unfortunately lacking a satisfying geometric interpretation. Nevertheless, we formulated precise conjectures describing their arithmetic significance which we were able to substantiate with both numerical and theoretical evidence.