Sandpile groups of supersingular isogeny graphs

Nathanaël Munier, Ari Shnidman

Research output: Contribution to journalArticlepeer-review

Abstract

Let p and q be distinct primes, and let Xp,qbe the (q + 1)- regular graph whose nodes are supersingular elliptic curves over Fpand 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 Xp,qas 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.

Original languageAmerican English
Pages (from-to)751-774
Number of pages24
JournalJournal de Theorie des Nombres de Bordeaux
Volume35
Issue number3
DOIs
StatePublished - 2023

Bibliographical note

Publisher Copyright:
© Les auteurs, 2023.

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