Sandpile groups of supersingular isogeny graphs

Let p and q be distinct primes, and let X p,q be the (q+1)-regular graph whose nodes are supersingular elliptic curves over 𝔽 ¯ p and whose edges are q-isogenies. For fixed p, we compute the distribution of the ℓ-Sylow subgroup of the sandpile group (i.e. Jacobian) of X p,q as q→∞. We find that the distribution disagrees with the Cohen–Lenstra heuristic in this context. Our proof is via Galois representations attached to modular curves. As a corollary, we give an upper bound on the probability that the Jacobian is cyclic, which we conjecture to be sharp.