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
Eyal Z. Goren 1; Payman L Kassaei 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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Eyal Z. Goren and Payman L Kassaei},
     title = {$p$-adic {Dynamics} of {Hecke} {Operators} on {Modular} {Curves}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {387--431},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {2},
     year = {2021},
     doi = {10.5802/jtnb.1165},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1165/}
AU  - Eyal Z. Goren
AU  - Payman L Kassaei
TI  - $p$-adic Dynamics of Hecke Operators on Modular Curves
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 387
EP  - 431
VL  - 33
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1165/
DO  - 10.5802/jtnb.1165
LA  - en
ID  - JTNB_2021__33_2_387_0
ER  - 
%0 Journal Article
%A Eyal Z. Goren
%A Payman L Kassaei
%T $p$-adic Dynamics of Hecke Operators on Modular Curves
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 387-431
%V 33
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1165/
%R 10.5802/jtnb.1165
%G en
%F JTNB_2021__33_2_387_0
Eyal Z. Goren; Payman L Kassaei. $p$-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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1165/

[1] Matthew H. Baker; Enrique González-Jiménez; Josep González; Bjorn Poonen Finiteness results for modular curves of genus at least 2, Am. J. Math., Volume 127 (2005) no. 6, pp. 1325-1387 | DOI | MR | Zbl

[2] Yves Benoist; Jean-François Quint Random walks on projective spaces, Compos. Math., Volume 150 (2014) no. 9, pp. 1579-1606 | DOI | MR | Zbl

[3] Yves Benoist; Jean-François Quint Random walks on reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 62, Springer, 2016 | MR | Zbl

[4] Pierre Berthelot Cohomologie rigide et cohomologie rigide à supports propres. Première partie (1996) (prépublication IRMAR 96-03, 89 pages, available from https://perso.univ-rennes1.fr/pierre.berthelot/)

[5] Ching-Li Chai Hecke orbits as Shimura varieties in positive characteristic, Proceedings of the international Congress of Mathematicians. Vol. II, European Mathematical Society, 2006, pp. 295-312 | Zbl

[6] Ching-Li Chai; Frans Oort Moduli of abelian varieties and p-divisible groups, Arithmetic geometry (Clay Mathematics Proceedings), Volume 8, American Mathematical Society, 2009, pp. 441-536 | MR | Zbl

[7] Ching-Li Chai; Frans Oort Monodromy and irreducibility of leaves, Ann. Math., Volume 173 (2011) no. 3, pp. 1359-1396 | DOI | MR | Zbl

[8] Laurent Clozel; Emmanuel Ullmo Équidistribution de sous-variétés spéciales, Ann. Math., Volume 161 (2005) no. 3, pp. 1571-1588 | DOI | Zbl

[9] Laurent Clozel; Emmanuel Ullmo :Équidistribution adélique des tores et équidistribution des points CM, Doc. Math., Volume Extra Vol. (2006), pp. 233-260 | Zbl

[10] Brian Conrad Several approaches to non-Archimedean geometry, p-adic geometry (University Lecture Series), Volume 45, American Mathematical Society, 2008, pp. 9-63 | MR | Zbl

[11] Daniel Disegni p-adic equidistribution of CM points (2019) (https://arxiv.org/abs/1904.07743)

[12] Manfred Einsiedler; Thomas Ward Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, 259, Springer, 2011 | MR | Zbl

[13] Noam Elkies; Ken Ono; Tonghai Yang Reduction of CM elliptic curves and modular function congruences, Int. Math. Res. Not., Volume 2005 (2005) no. 44, pp. 2695-2707 | DOI | MR | Zbl

[14] Eyal Z. Goren; Payman L. Kassaei Canonical subgroups over Hilbert modular varieties, J. Reine Angew. Math., Volume 670 (2012), pp. 1-63 | DOI | MR | Zbl

[15] Benedict H. Gross Heights and the special values of L-series, Number theory (Montreal, Quebec, 1985) (CMS Conference Proceedings), Volume 7, American Mathematical Society, 1985, pp. 115-187 | Zbl

[16] Benedict H. Gross On canonical and quasicanonical liftings, Invent. Math., Volume 84 (1986) no. 2, pp. 321-326 | DOI | MR | Zbl

[17] Michiel Hazewinkel Formal groups and applications, AMS Chelsea Publishing, 2012 (corrected reprint of the 1978 original) | DOI | Zbl

[18] Sebastián Herrero; Ricard Menares; Juan Rivera-Letelier p-adic distribution of CM points and Hecke orbits. II. Linnik equidistribution on the supersingular locus (to appear)

[19] Sebastián Herrero; Ricard Menares; Juan Rivera-Letelier p-adic distribution of CM points and Hecke orbits. I: Convergence towards the Gauss point, Algebra Number Theory, Volume 14 (2020) no. 5, pp. 1239-1290 | DOI | MR | Zbl

[20] Michael J. Hopkins; Benedict H. Gross Equivariant vector bundles on the Lubin-Tate moduli space, Topology and representation theory (Evanston, IL, 1992) (Contemporary Mathematics), Volume 158, American Mathematical Society, 1992, pp. 23-88 | DOI | MR | Zbl

[21] Nicholas Katz Serre–Tate local moduli, Algebraic surfaces (Orsay, 1976–78) (Lecture Notes in Mathematics), Volume 868, Springer, 1981, p. 1976-78 | MR | Zbl

[22] Jonathan Lubin; John Tate Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. Fr., Volume 94 (1966), pp. 49-59 | DOI | Numdam | MR | Zbl

[23] William Messing The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, 264, Springer, 1972 | MR | Zbl

[24] Frans Oort Foliations in moduli spaces of abelian varieties and dimension of leaves, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II (Progress in Mathematics), Volume 270, Birkhäuser, 2009, pp. 465-501 | DOI | MR | Zbl

[25] Naser T. Sardari Optimal strong approximation for quadratic forms, Duke Math. J., Volume 168 (2019) no. 10, pp. 1887-1927 | MR | Zbl

[26] Stanley Sawyer Isotropic random walks in a tree, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 42 (1978) no. 4, pp. 279-292 | DOI | MR | Zbl

[27] Jean-Pierre Serre Two letters on quaternions and modular forms (modp), Isr. J. Math., Volume 95 (1996), pp. 281-299 | DOI | Zbl

[28] Joseph H. Silverman The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, 2009 | MR | Zbl

[29] Barry Simon Convexity. An analytic viewpoint, Cambridge Tracts in Mathematics, 187, Cambridge University Press, 2011 | Zbl

[30] Andrew V. Sutherland Isogeny volcanoes, ANTS X–Proceedings of the Tenth Algorithmic Number Theory Symposium (The Open Book Series), Volume 1, Mathematical Sciences Publishers, 2013, pp. 507-530 | MR | Zbl

[31] Wolfgang Woess Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, 138, Cambridge University Press, 2000 | MR | Zbl

Cited by Sources: