$p$-adic Dynamics of Hecke Operators on Modular Curves

Goren, Eyal Z.; Kassaei, Payman L

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 2, pp. 387-431.

Cet article se penche sur la dynamique $p$-adique des opérateurs de Hecke d’indice premier à $p$ agissant sur les points des courbes modulaires dans les cas de bonne réduction ordinaire et supersingulière. Une attention particulière est accordée à la dynamique des points CM. Dans le cas de réduction ordinaire, nous exploitons les coordonnées de Serre–Tate, alors que dans le cas de réduction supersingulière nous utilisons un paramètre sur l’espace de déformations de l’unique groupe formel de hauteur $2$ sur ${\overline{𝔽}}_p$ et le morphisme de périodes de Gross–Hopkins.

In this paper we study the $p$-adic dynamics of prime-to-$p$ Hecke operators on the set of points of modular curves in both cases of good ordinary and supersingular reduction. We pay special attention to the dynamics on the set of CM points. In the case of ordinary reduction we employ the Serre–Tate coordinates, while in the case of supersingular reduction we use a parameter on the deformation space of the unique formal group of height $2$ over ${\overline{𝔽}}_p$, and take advantage of the Gross–Hopkins period map.

Classification : 11F32, 11G18, 11G15, 14G35
Mots clés : $p$-adic Dynamics, Hecke operators, Modular curves, Serre–Tate coordinates, Gross–Hopkins period map

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