STNB2024(37a edició)
Plectic insights on the BSD conjecture for higher rank elliptic curves
Coordinación
Michele Fornea (CRM)
Descripción
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 and Molina 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 lecture series 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.
Charlas
- Gauss' class number problem and the BSD conjecture (Michele Fornea)
- Introduction to the plectic philosophy (Michele Fornea)
- Bestiary of plectic points (Michele Fornea)
- Iwasawa theory and mock plectic points (Michele Fornea)
