Automorphisms of the supersingular isogeny graph (Q6185634)

Fix two distinct primes \(p,\ell\in\mathbb{Z}\) and let \(\mathscr{G}(p,\ell)\) be the \(\ell\)-isogeny graph of (isomorphism classes of) supersingular elliptic curves over \(\overline{\mathbb{F}_p}\). The graph \(\mathscr{G}(p,\ell)\) has relevant applications in cryptography, and the paper deals with its automorphism group \(\text{Aut}^*(\mathscr{G}(p,\ell))\) (where one mods out the subgroup of \(\text{Aut}(\mathscr{G}(p,\ell))\) fixing every vertex), which provides a measure of the complexity of the \textit{path finding problem} on \(\mathscr{G}(p,\ell)\). Looking at the action on the free group of vertices (i.e., on classes of supersingular curves), the author is able to embed \(\text{Aut}^*(\mathscr{G}(p,\ell))\) into the Hecke algebra \((\mathbb{T}^*)_{tor}/(\pm 1)\) whenever \(T_\ell\) generates the whole algebra \(\mathbb{T}_\mathbb{C}\) over \(\mathbb{C}\). Then, from the action of \(\mathbb{T}\) on \(X_0(p)\), one derives an embedding \[ \text{Aut}^*(\mathscr{G}(p,\ell)) \hookrightarrow \text{Aut}_\mathbb{C}(X_0(p))\,. \] The latter group is known to be of order 2 for all \(p>37\) with nontrivial element provided by the Atkin-Lehner involution acting as the Frobenius on the supersingular points of \(X_0(p)_{\mathbb{F}_p}\). Applying a result of \textit{A. P. Ogg} [Semin. Delange-Pisot-Poitou, Theorie des Nombres, Fasc. 1, Expose 7, (1975; Zbl 0336.14006)], the author shows that \(\text{Aut}^*(\mathscr{G}(p,\ell))\simeq \mathbb{Z}/(2)\mathbb{Z}\) for a density 1 set of primes \(p>71\).