Isogeny volcanoes and cryptography

Elliptic curves are rich geometric objects with interesting properties depending on their field of definition. In the case where these curves are defined over a finite field, we consider special maps between these curves: the isogenies, which are surjective morphisms with finite kernel. Imagine a graph in which vertices represent elliptic curves and edges represent isogenies of a fixed degree… the goal of the talk is to describe this setting, and to explain what kind of structure these graphs have… in the ordinary case, very volcanic! We will also explain a link with modern cryptography, and we will conclude with mentioning a new result, obtained in collaboration with Francesco Campagna and Henry Bambury, concerning a related inverse problem.