A geometric view on Iwasawa theory

Betina, Adel; Dimitrov, Mladen

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

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 $f$.

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

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

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

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

[4] Bellaïche, Joël; Chenevier, Gaëtan Lissité de la courbe de Hecke de ${\mathrm{GL}}_2$ aux points Eisenstein critiques, J. Inst. Math. Jussieu, Volume 5 (2006) no. 2, pp. 333-349 | DOI

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

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

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

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

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

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

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

[13] Breuil, Christophe; Emerton, Matthew Représentations $p$-adiques ordinaires de ${\mathrm{GL}}_2\left({\mathbf{Q}}_p\right)$ 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] Buzzard, Kevin Eigenvarieties, $L$-functions and Galois representations (London Mathematical Society Lecture Note Series), Volume 320, Cambridge University Press, 2007, pp. 59-120 | DOI | MR

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[36] Hida, Haruzo 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] Hida, Haruzo 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] Hida, Haruzo Galois representations into ${\mathrm{GL}}_2\left({\mathbf{Z}}_p\left[\left[X\right]\right]\right)$ attached to ordinary cusp forms, Invent. Math., Volume 85 (1986), pp. 545-613 | DOI | MR

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

[40] Hsu, Chi-Yun 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] Lee, Hao Irregular weight one points with $D_4$ image, Can. Math. Bull., Volume 62 (2019) no. 1, pp. 109-118 | DOI | MR | Zbl

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

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

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

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