p-adic Dynamics of Hecke Operators on Modular Curves
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 387-431.

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 𝔽 ¯ p , and take advantage of the Gross–Hopkins period map.

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 𝔽 ¯ p et le morphisme de périodes de Gross–Hopkins.

Published online:
DOI: 10.5802/jtnb.1165
Classification: 11F32, 11G18, 11G15, 14G35
Keywords: $p$-adic Dynamics, Hecke operators, Modular curves, Serre–Tate coordinates, Gross–Hopkins period map
Goren, Eyal Z. 1; Kassaei, Payman L 2

1 Department of Mathematics and Statistics McGill University 805 Sherbrooke St. W. Montreal H3A 0B9, QC, Canada
2 Department of Mathematics King’s College London Strand, London WC2R 2LS, United Kingdom
     author = {Goren, Eyal Z. and Kassaei, Payman L},
     title = {$p$-adic {Dynamics} of {Hecke} {Operators} on {Modular} {Curves}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Goren, Eyal Z.; Kassaei, Payman L. $p$-adic Dynamics of Hecke Operators on Modular Curves. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 387-431. doi : 10.5802/jtnb.1165. http://archive.numdam.org/articles/10.5802/jtnb.1165/

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