Isogeny graphs and Isogeny Volcanoes

Abstract

\textit{Isogeny graphs} are a type of graphs, where the vertices represent elliptic curves and the edges represent isogenies. I will examine some of the structures of these graphs in this thesis. It turns out that the majority of the components of such a graph will be \textit{volcanoes}, see \cref{defn:pvolcano}. This has applications in cryptography and number theory, because many algorithms are made more efficient by exploiting this structure. In most elliptic curve cryptography one is dependent on computing an elliptic curve with a given number of points over a fixed field. The \textit{complex multiplication method} in \cref{rmk:CMmethod} uses the volcano structure to compute such an elliptic curve.