Sandpile groups of supersingular isogeny graphs

Nathanaël Munier; Ari Shnidman

doi:10.5802/jtnb.1262

Nathanaël Munier ¹ ; Ari Shnidman ²

¹ Institut de Mathématiques de Toulouse 1 R.3 Université Paul Sabatier 118 Rte de Narbonne 31400 Toulouse, France
² Einstein Institute of Mathematics The Hebrew University of Jerusalem Edmund J. Safra Campus Jerusalem 9190401, Israel

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 751-774.

Résumé
Abstract

Soient $p$ et $q$ deux nombres premiers distincts, et soit $X_{p, q}$ le graphe $(q + 1)$ -régulier dont les nœuds sont les courbes elliptiques supersingulières sur ${\bar{𝔽}}_{p}$ et dont les arêtes sont les $q$ -isogénies. Pour une valeur de $p$ fixée, la distribution des sous-groupes de $ℓ$ -Sylow du groupe jacobien de $X_{p, q}$ est donnée pour $q \to \infty$ . Nous constatons que cette distribution ne correspond pas à l’heuristique de Cohen–Lenstra dans ce contexte. La preuve que nous donnons utilise des représentations de Galois reliées à des courbes modulaires. Comme corollaire, nous donnons une borne supérieure sur la probabilité que le groupe jacobien soit cyclique, que nous conjecturons être optimale.

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 ${\bar{𝔽}}_{p}$ 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 $q \to \infty$ . 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, we give an upper bound on the probability that the Jacobian is cyclic, which we conjecture to be sharp.

Reçu le : 2022-05-21
Accepté le : 2023-01-21
Publié le : 2024-03-04

DOI : 10.5802/jtnb.1262

Classification : 05C48, 11R29
Mots clés : Graphs, Jacobians, isogenies, modular curves

Affiliations des auteurs :

Nathanaël Munier ¹ ; Ari Shnidman ²

¹ Institut de Mathématiques de Toulouse 1 R.3 Université Paul Sabatier 118 Rte de Narbonne 31400 Toulouse, France
² Einstein Institute of Mathematics The Hebrew University of Jerusalem Edmund J. Safra Campus Jerusalem 9190401, Israel

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Sandpile groups of supersingular isogeny graphs},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {751--774},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Nathanaël Munier; Ari Shnidman. Sandpile groups of supersingular isogeny graphs. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 751-774. doi : 10.5802/jtnb.1262. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1262/

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