Isogeny graphs with level structures arrising from the Verschiebung map

TL;DR

This paper studies enhanced isogeny graphs of elliptic curves with level structures from the Verschiebung map, revealing their tower and volcanic structures, thus advancing understanding of their geometric and algebraic properties.

Contribution

It introduces a novel class of isogeny graphs with level structures from the Verschiebung map and proves their tower and volcanic properties.

Findings

Graphs form $\\mathbb{Z}_p$-towers of coverings.

Connected components exhibit volcanic structure.

Extends previous results to new level structures.

Abstract

We enhance an isogeny graph of elliptic curves by incorporating level structures defined by bases of the kernels of iterates of the Verschiebung map. We extend several previous results on isogeny graphs with level structures defined by geometric points to these graphs. Firstly, we prove that these graphs form $\mathbb{Z}_p$-towers of graph coverings as the power of the Verschiebung map varies. Secondly, we prove that the connected components of these graphs display a volcanic structure.