Sandpile groups of supersingular isogeny graphs

TL;DR

This paper investigates the distribution of the Jacobian's Sylow subgroups of supersingular isogeny graphs, revealing deviations from Cohen-Lenstra heuristics and employing Galois representations for proofs.

Contribution

It provides the first detailed analysis of the Jacobian's Sylow subgroup distribution in supersingular isogeny graphs, challenging existing heuristics.

Findings

Distribution of Sylow subgroups differs from Cohen-Lenstra predictions

Provides an upper bound on the Jacobian's cyclic probability

Uses Galois representations attached to modular curves for proofs

Abstract

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 $\overline{\mathbb{F}}_p$ and whose edges are $q$-isogenies. For fixed $p$, we compute the distribution of the $\ell$-Sylow subgroup of the sandpile group (i.e.\ Jacobian) of $X_{p,q}$ as $q \to \infty$. 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 of our result, we give an upper bound on the probability that the Jacobian is cyclic, which we conjecture to be sharp.