On towers of isogeny graphs with full level structures

Let p and \(\ell \) $\ell$ be distinct prime numbers, let q be a power of a prime number r that is distinct from p and \(\ell \) $\ell$ , and let M be a positive integer coprime to \(q\ell \) $q\ell$ . We define the directed graph \(X_\ell ^q(M)\) $X_{\ell }^q(M)$ whose vertices are given by isomorphism classes of elliptic curves over the finite field of q elements enhanced with the full level M structure. The edges of \(X_\ell ^q(M)\) $X_{\ell }^q(M)$ are given by \(\ell \) $\ell$ -isogenies. Fix a positive integer N, and write \(M=p^nN\) $M=p^{n}N$ . We are interested in when the connected components of \(X_\ell ^q(p^nN)\) $X_{\ell }^q(p^{n}N)$ give rise to a tower of Galois coverings as n varies. We analyze the structure of the inverse limit of the Galois groups of these coverings as a p-adic Lie group. We also study similar towers of isogeny graphs given by oriented elliptic curves enhanced with a full level structure.