Abstract

Fifteen years ago Henri Darmon introduced a construction of p-adic points on elliptic curves. These points were conjectured to be algebraic and to behave much like Heegner points, although so far a proof remains inaccessible. Other constructions emerged in the subsequent years, thanks to work of himself as well as Adam Logan, Matthew Greenberg, Mak Trifkovic and Jerome Gartner, among others. Some of these constructions were also non-archimedean, but others construct points which a priori are defined over the complex numbers. So far none of these constructions are known to yield algebraic points, although there is an extensive and beautiful collection of numerical evidence.

In this talk I will present joint work with Xavier Guitart and Haluk Sengun, in which we propose a framework that includes all the above constructions as particular cases, and which allows us to extend the construction of local points to elliptic curves defined over arbitrary number fields. I will explain our construction and present numerical evidence supporting the new cases.