On ordinary isogeny graphs with level structures

Abstract

Let $\ell$ and $p$ be two distinct prime numbers. We study $\ell$-isogeny graphs of ordinary elliptic curves defined over a finite field of characteristic $p$, together with a level structure. Firstly, we show that as the level varies over all $p$-powers, the graphs form an Iwasawa-theoretic abelian $p$-tower, which can be regarded as a graph-theoretical analogue of the Igusa tower of modular curves. Secondly, we study the structure of the crater of these graphs, generalizing previous results on volcano graphs. Finally, we solve an inverse problem of graphs arising from the crater of $\ell$-isogeny graphs with level structures, partially generalizing a recent result of Bambury, Campagna and Pazuki.