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arXiv · 2111.10389

Sandpile groups of supersingular isogeny graphs

Nathanaël Munier·Ari Shnidman
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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.

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Nathanaël Munier, Ari Shnidman. 2021-11-19. Sandpile groups of supersingular isogeny graphs. https://arxiv.org/abs/2111.10389

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