A geometric view on Iwasawa theory

Adel Betina; Mladen Dimitrov

doi:10.5802/jtnb.1176

Adel Betina ¹ ; Mladen Dimitrov ²

¹ University of Vienna, Faculty of Mathematics Oskar-Morgenstern-Platz 1 1090 Wien, Austria
² University of Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé 59000 Lille, France

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.1, pp. 703-731.

Résumé
Abstract

Cet article prolonge notre étude de la géométrie de la courbe $p$ -adique de Hecke en un point défini par une forme modulaire cuspidale $f$ de poids $1$ à multiplication complexe et irrégulière en $p$ , et des implications en théories d’Iwasawa et de Hida. Les nouveaux résultats incluent la détermination des coefficients de Fourier de certaines formes modulaires $p$ -adiques non-classiques appartenant à l’espace propre généralisé de $f$ , en termes de logarithmes $p$ -adiques de nombres algébriques. Nous calculons aussi le « mystérieux » bi-rapport des filtrations $p$ -ordinaires des familles de Hida contenant $f$ .

This article extends our study of the geometry of the $p$ -adic eigencurve at a point defined by a weight $1$ cuspform $f$ irregular at $p$ and having complex multiplication, and the implications in Iwasawa and in Hida theories. The novel results include the determination of the Fourier coefficients of certain non-classical $p$ -adic modular forms belonging to the generalized eigenspace of $f$ , in terms of $p$ -adic logarithms of algebraic numbers. We also compute the “mysterious” cross-ratios of the $p$ -ordinary filtrations of the Hida families containing $f$ .

Reçu le : 2020-03-28
Accepté le : 2020-11-04
Publié le : 2022-01-26

DOI : 10.5802/jtnb.1176

Classification : 11F33, 11R23, 11F80
Mots clés : Hida family, weight one modular form, eigencurve, $p$-adic $L$-function

Affiliations des auteurs :

Adel Betina ¹ ; Mladen Dimitrov ²

¹ University of Vienna, Faculty of Mathematics Oskar-Morgenstern-Platz 1 1090 Wien, Austria
² University of Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé 59000 Lille, France

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A geometric view on {Iwasawa} theory},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {703--731},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {3.1},
     year = {2021},
     doi = {10.5802/jtnb.1176},
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Adel Betina; Mladen Dimitrov. A geometric view on Iwasawa theory. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.1, pp. 703-731. doi : 10.5802/jtnb.1176. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1176/

Bibliographie
Cité par

[1] Joël Bellaïche $p$ -adic $L$ -functions of critical CM forms (2011) (preprint)

[2] Joël Bellaïche Critical $p$ -adic $L$ -functions, Invent. Math., Volume 189 (2012) no. 1, pp. 1-60 | DOI | MR | Zbl

[3] Joël Bellaïche The eigenbook. Eigenvarieties, families of Galois representations, $p$ -adic $L$ -functions, Pathways in Mathematics, Birkhäuser, 2021, xi+316 pages | DOI

[4] Joël Bellaïche; Gaëtan Chenevier Lissité de la courbe de Hecke de ${GL}_{2}$ aux points Eisenstein critiques, J. Inst. Math. Jussieu, Volume 5 (2006) no. 2, pp. 333-349 | DOI

[5] Joël Bellaïche; Gaëtan Chenevier Families of Galois representations and Selmer groups, Astérisque, Société Mathématique de France, 2009, xii+314 pages | Numdam

[6] Joël Bellaïche; Mladen Dimitrov On the eigencurve at classical weight 1 points, Duke Math. J., Volume 165 (2016) no. 2, pp. 245-266 | MR | Zbl

[7] John Bergdall Ordinary modular forms and companion points on the eigencurve, J. Number Theory, Volume 134 (2014), pp. 226-239 | DOI | MR

[8] Adel Betina Ramification of the eigencurve at classical RM points, Can. J. Math., Volume 72 (2020) no. 1, pp. 57-88 | DOI | MR

[9] Adel Betina; Mladen Dimitrov Geometry of the eigencurve at CM points and trivial zeros of Katz $p$ -adic $L$ -functions, Adv. Math., Volume 384 (2021), 107724, 43 pages | MR

[10] Adel Betina; Mladen Dimitrov; Alice Pozzi On the failure of gorensteinness at weight 1 Eisenstein points of the eigencurve, Am. J. Math., Volume 144 (2022) no. 1 (34 pages)

[11] Adel Betina; Mladen Dimitrov; S.-C. Shih Eisenstein points on the Hilbert cuspidal eigenvariety (2020) (preprint)

[12] Adel Betina; Chris Williams Arithmetic of $p$ -irregular modular forms: families and $p$ -adic $L$ -functions, Mathematika, Volume 67 (2021) no. 4, pp. 917-948 | DOI | MR

[13] Christophe Breuil; Matthew Emerton Représentations $p$ -adiques ordinaires de ${GL}_{2} (Q_{p})$ et compatibilité local-global, Représentations $p$ -adiques de groupes $p$ -adiques III: Méthodes globales et géométriques (Astérisque), Volume 331, Société Mathématique de France, 2010, pp. 255-315 | Numdam | Zbl

[14] Kevin Buzzard Eigenvarieties, $L$ -functions and Galois representations (London Mathematical Society Lecture Note Series), Volume 320, Cambridge University Press, 2007, pp. 59-120 | DOI | MR

[15] Frank Calegari; Matthew Emerton On the ramification of Hecke algebras at Eisenstein primes, Invent. Math., Volume 160 (2005) no. 1, pp. 97-144 | DOI | MR

[16] Francesc Castella; Carl Wang-Erickson; Haruzo Hida Class groups and local indecomposability for non-CM forms, J. Eur. Math. Soc. (2021) (published online first) | DOI

[17] S. Cho; Vinayak Vatsal Deformations of induced Galois representations, J. Reine Angew. Math., Volume 556 (2003), pp. 79-98 | MR

[18] Robert F. Coleman Classical and overconvergent modular forms, Invent. Math., Volume 124 (1996) no. 1-3, pp. 215-241 | DOI | MR

[19] Robert F. Coleman; Bas Edixhoven On the semi-simplicity of the $U_{p}$ -operator on modular forms, Math. Ann., Volume 310 (1998) no. 1, pp. 119-127 | DOI | MR

[20] Robert F. Coleman; Barry Mazur The eigencurve, Galois representations in arithmetic algebraic geometry (London Mathematical Society Lecture Note Series), Volume 254, Cambridge University Press, 1996, pp. 1-113

[21] Henri Darmon; Alan Lauder; Victor Rotger Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields, Adv. Math., Volume 283 (2015), pp. 130-142 | DOI | MR

[22] Henri Darmon; Alan Lauder; Victor Rotger First order $p$ -adic deformations of weight one newforms, $L$ -functions and automorphic forms (Contributions in Mathematical and Computational Sciences), Volume 10, Springer, 2017, pp. 39-80 | DOI | MR

[23] Samit Dasgupta; Henri Darmon; Robert Pollack Hilbert modular forms and the Gross–Stark conjecture, Ann. Math., Volume 174 (2011) no. 1, pp. 439-484 | DOI | MR

[24] Samit Dasgupta; Mahesh Kakde; Kevin Ventullo On the Gross–Stark conjecture, Ann. Math., Volume 188 (2018) no. 3, pp. 833-870 | MR

[25] Pierre Deligne; Jean-Pierre Serre Formes modulaires de poids $1$ , Ann. Sci. Éc. Norm. Supér., Volume 7 (1974), pp. 507-530 | DOI | Numdam | MR

[26] Hansheng Diao; Ruochuan Liu The eigencurve is proper, Duke Math. J., Volume 165 (2016) no. 7, pp. 1381-1395 | MR | Zbl

[27] Mladen Dimitrov On the local structure of ordinary Hecke algebras at classical weight one points, Automorphic forms and Galois representations (London Mathematical Society Lecture Note Series), Volume 415, Cambridge University Press, 2014, pp. 1-16 | MR

[28] Mladen Dimitrov; Eknath Ghate On classical weight one forms in Hida families, J. Théor. Nombres Bordeaux, Volume 24 (2012) no. 3, pp. 669-690 | DOI | Numdam | MR | Zbl

[29] Bruce Ferrero; Ralph Greenberg On the behavior of $p$ -adic $L$ -functions at $s = 0$ , Invent. Math., Volume 50 (1978), pp. 91-102 | DOI | MR

[30] Eknath Ghate On the local behavior of ordinary modular Galois representations, Modular curves and Abelian varieties (Progress in Mathematics), Volume 224, Birkhäuser, 2004, pp. 105-124 | DOI | MR | Zbl

[31] Eknath Ghate; Narasimha Kumar Control theorems for ordinary $2$ -adic families of modular forms, Automorphic representations and $L$ -functions (Tata Institute of Fundamental Research Studies in Mathematics), Volume 22, Tata Institute of Fundamental Research, 2013, pp. 231-261 | MR | Zbl

[32] Eknath Ghate; Vinayak Vatsal On the local behaviour of ordinary $Λ$ -adic representations, Ann. Inst. Fourier, Volume 54 (2004) no. 7, pp. 2143-2162 | DOI | MR | Zbl

[33] Ralph Greenberg; Glenn Stevens $p$ -adic $L$ -functions and $p$ -adic periods of modular forms, Invent. Math., Volume 111 (1993) no. 2, pp. 407-447 | DOI | MR

[34] David Hansen Universal eigenvarieties, trianguline Galois representations, and $p$ -adic Langlands functoriality, J. Reine Angew. Math., Volume 730 (2017), pp. 1-64 (With an appendix by James Newton) | DOI | MR

[35] Haruzo Hida Congruence of cusp forms and special values of their zeta functions, Invent. Math., Volume 63 (1981), pp. 225-261 | DOI | MR

[36] Haruzo Hida On congruence divisors of cusp forms as factors of the special values of their zeta functions, Invent. Math., Volume 64 (1981), pp. 221-262 | DOI | MR

[37] Haruzo Hida Kummer’s criterion for the special values of Hecke $L$ -functions of imaginary quadratic fields and congruences among cusp forms, Invent. Math., Volume 66 (1982), pp. 415-459 | DOI | MR

[38] Haruzo Hida Galois representations into ${GL}_{2} (Z_{p} [[X]])$ attached to ordinary cusp forms, Invent. Math., Volume 85 (1986), pp. 545-613 | DOI | MR

[39] Haruzo Hida Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Éc. Norm. Supér., Volume 19 (1986), pp. 231-273 | DOI | Numdam | MR

[40] Chi-Yun Hsu Fourier coefficients of the overconvergent generalized eigenform associated to a CM form, Int. J. Number Theory, Volume 16 (2020) no. 6, pp. 1185-1197 | MR

[41] Hao Lee Irregular weight one points with $D_{4}$ image, Can. Math. Bull., Volume 62 (2019) no. 1, pp. 109-118 | DOI | MR | Zbl

[42] Dipramit Majumdar Geometry of the eigencurve at critical Eisenstein series of weight 2, J. Théor. Nombres Bordeaux, Volume 27 (2015) no. 1, pp. 183-197 | DOI | Numdam | MR

[43] Barry Mazur; John Tate; Jeremy Teitelbaum On $p$ -adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math., Volume 84 (1986), pp. 1-48 | DOI | MR

[44] Vincent Pilloni Overconvergent modular forms, Ann. Inst. Fourier, Volume 63 (2013) no. 1, pp. 219-239 | DOI | Numdam | MR

[45] Jean-Pierre Serre Formes modulaires et fonctions zêta $p$ -adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) (Lecture Notes in Mathematics), Volume 350, Springer, 1972, pp. 191-268 | DOI

[46] Preston Wake; Carl Wang-Erickson Pseudo-modularity and Iwasawa theory, Am. J. Math., Volume 140 (2018) no. 4, pp. 977-1040 | DOI | MR

[47] Andrew Wiles The Iwasawa conjecture for totally real fields, Ann. Math., Volume 131 (1990) no. 3, pp. 493-540 | DOI | MR

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