Sandpile groups of supersingular isogeny graphs

2021/11/19 by Munier, Nathanaël, Shnidman, Ari
#05C48 #11R29 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2111.10389

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 \mathbbF_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 of our result, we give an upper bound on the probability that the Jacobian is cyclic, which we conjecture to be sharp.

Related

• Higher-degree supersingular group actions
• The Distribution of Sandpile Groups of Random Graphs with their Pairings
• The distribution of sandpile groups of random graphs
• The Spine of a Supersingular ℓ-Isogeny graph
• Semisimple pointed isogeny graphs for abelian varieties