Congruences of modular forms and the Iwasawa $\lambda $-invariants

Hirano, Yuichi

doi:10.24033/bsmf.2752

Congruences of modular forms and the Iwasawa

λ

-invariants
[Congruences de formes modulaires et

λ

-invariants d’Iwasawa]

Hirano, Yuichi ¹

¹ Graduate School of Mathematical Sciences, The University of Tokyo, 8-1 Komaba 3-chome, Meguro-ku, Tokyo, 153-8914, Japan

Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 1, pp. 1-79

Abstract (VO)
Résumé

In this paper, we show how congruences between cusp forms and Eisenstein series of weight $k \geq 2$ give rise to corresponding congruences between the algebraic parts of the critical values of the associated $L$ -functions. This is a generalization of results of Mazur, Stevens, and Vatsal in the case where $k = 2$ . As an application, by proving congruences between the $p$ -adic $L$ -function of a certain cusp form and the product of two Kubota-Leopoldt $p$ -adic $L$ -functions, we prove the Iwasawa main conjecture (up to $p$ -power) for cusp forms at ordinary primes $p$ when the associated residual Galois representations are reducible. This is a generalization of Greenberg and Vatsal in the case where $k = 2$ .

Dans cet article, nous montrons comment les congruences entre formes paraboliques et séries d’Eisenstein de poids $k \geq 2$ donnent lieu à des congruences entre les parties algébriques des valeurs critiques des fonctions $L$ associées. C’est une généralisation des travaux de Mazur, Stevens et Vatsal dans le cas où $k = 2$ . Comme application, en prouvant des congruences entre la fonction $p$ -adique $L$ d’une certaine forme parabolique et le produit de deux fonctions de Kubota-Leopoldt $p$ -adiques $L$ , nous prouvons la conjecture principale d’Iwasawa (à puissance $p$ près) pour les formes paraboliques à nombres premiers ordinaires $p$ lorsque les représentations de Galois résiduelles associées sont réductibles. C’est une généralisation des travaux de Greenberg et Vatsal dans le cas où $k = 2$ .

MR Zbl

DOI : 10.24033/bsmf.2752

Classification : 11F33, 11F67, 11F75, 11R23, 14F30
Keywords: Congruences of modular forms, special values of $L$-functions, Iwasawa theory, Eisenstein series, Mellin transform, $p$-adic Hodge theory

Affiliations des auteurs :

Hirano, Yuichi ¹

¹ Graduate School of Mathematical Sciences, The University of Tokyo, 8-1 Komaba 3-chome, Meguro-ku, Tokyo, 153-8914, Japan

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     title = {Congruences of modular forms and the {Iwasawa} $\lambda $-invariants},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {1--79},
     year = {2018},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {146},
     number = {1},
     doi = {10.24033/bsmf.2752},
     mrnumber = {3864870},
     zbl = {1446.11076},
     language = {en},
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Hirano, Yuichi. Congruences of modular forms and the Iwasawa $\lambda $-invariants. Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 1, pp. 1-79. doi: 10.24033/bsmf.2752

Bibliographie
Cité par

Ash, Avner; Stevens, Glenn Modular forms in characteristic $ℓ$ and special values of their $L$ -functions, Duke Math. J., Volume 53 (1986), pp. 849-868 | DOI | MR | Zbl

Amice, Yvette; Vélu, Jacques Distributions $p$ -adiques associées aux séries de Hecke, Astérisque, Volume 24–25 (1975), pp. 119-131 | MR | Zbl | Numdam

Berger, T. An Eisenstein ideal for imaginary quadratic fields, Ph. D. Thesis , University of Michigan (2005) | MR

Breuil, Christophe; Messing, William Torsion étale and crystalline cohomologies, Astérisque, Volume 279 (2002), pp. 81-124 | MR | Zbl | Numdam

Borel, Armand Stable real cohomology of arithmetic groups. II, Manifolds and Lie groups (Notre Dame, Ind., 1980) (Progr. Math.), Volume 14, Birkhäuser, 1981, pp. 21-55 | MR | Zbl | DOI

Breuil, Christophe Cohomologie étale de $p$ -torsion et cohomologie cristalline en réduction semi-stable, Duke Math. J., Volume 95 (1998), pp. 523-620 | DOI | MR | Zbl

Deligne, Pierre Formes modulaires et représentations $l$ -adiques (Séminaire Bourbaki, vol. 1968/1969, exposé no 355, Lecture Notes in Math.), Volume 175, Springer, Berlin, 1971, pp. 139-172 | MR | Zbl | Numdam

Deligne, Pierre; Rapoport, M. Les schémas de modules de courbes elliptiques, Springer Lecture Notes in Math., Volume 349 (1973), pp. 143-316 | MR | Zbl | DOI

Faltings, Gerd Crystalline cohomology and $p$ -adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25-80 | MR | Zbl

Faltings, Gerd; Jordan, Bruce W. Crystalline cohomology and $GL (2, Q)$ , Israel J. Math., Volume 90 (1995), pp. 1-66 | DOI | MR | Zbl

Fontaine, Jean-Marc; Laffaille, Guy Construction de représentations $p$ -adiques, Ann. Sci. École Norm. Sup., Volume 15 (1982), pp. 547-608 | Numdam | MR | Zbl | DOI

Fontaine, Jean-Marc; Messing, William $p$ -adic periods and $p$ -adic étale cohomology, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) (Contemp. Math.), Volume 67, Amer. Math. Soc., Providence, RI, 1987, pp. 179-207 | DOI | MR | Zbl

Friedman, Eduardo C. Ideal class groups in basic $Z_{p_{1}} \times \dots \times Z_{p_{s}}$ -extensions of abelian number fields, Invent. math., Volume 65 (1981/82), pp. 425-440 | DOI | MR | Zbl

Ferrero, Bruce; Washington, Lawrence C. The Iwasawa invariant $μ_{p}$ vanishes for abelian number fields, Ann. of Math., Volume 109 (1979), pp. 377-395 | DOI | MR | Zbl

Greenberg, Ralph Iwasawa theory for $p$ -adic representations, Algebraic number theory (Adv. Stud. Pure Math.), Volume 17, Academic Press, Boston, MA, 1989, pp. 97-137 | MR | Zbl | DOI

Greenberg, Ralph; Vatsal, Vinayak On the Iwasawa invariants of elliptic curves, Invent. math., Volume 142 (2000), pp. 17-63 | DOI | MR | Zbl

Hida, Haruzo Elementary theory of $L$ -functions and Eisenstein series, London Mathematical Society Student Texts, 26, Cambridge Univ. Press, Cambridge, 1993, 386 pages | DOI | MR | Zbl

Heumann, Jay; Vatsal, Vinayak Modular symbols, Eisenstein series, and congruences, J. Théor. Nombres Bordeaux, Volume 26 (2014), pp. 709-757 | Numdam | MR | Zbl | DOI

Kato, Kazuya $p$ -adic Hodge theory and values of zeta functions of modular forms, Astérisque, Volume 295 (2004), pp. 117-290 | MR | Zbl | Numdam

Katz, Nicholas M. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Publ. Math. IHÉS, Volume 39 (1970), pp. 175-232 | Numdam | MR | Zbl | DOI

Katz, Nicholas M. $p$ -adic properties of modular schemes and modular forms, Springer Lecture Notes in Math., Volume 350 (1973), pp. 69-190 | MR | Zbl | DOI

Kato, Kazuya Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191-224 | MR | Zbl

Kato, Kazuya Toric singularities, Amer. J. Math., Volume 116 (1994), pp. 1073-1099 | DOI | MR | Zbl

Katz, Nicholas M.; Mazur, Barry Arithmetic moduli of elliptic curves, Annals of Math. Studies, 108, Princeton Univ. Press, Princeton, NJ, 1985, 514 pages | DOI | MR | Zbl

Mazur, Barry On the arithmetic of special values of $L$ functions, Invent. math., Volume 55 (1979), pp. 207-240 | DOI | MR | Zbl

Miyake, Toshitsune Modular forms, Springer Monographs in Math., Springer, Berlin, 2006, 335 pages | MR | Zbl

Mazur, Barry; Tate, J.; Teitelbaum, J. On $p$ -adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. math., Volume 84 (1986), pp. 1-48 | DOI | MR | Zbl

Mazur, Barry; Wiles, A. Class fields of abelian extensions of $Q$ , Invent. math., Volume 76 (1984), pp. 179-330 | DOI | MR | Zbl

Scholl, A. J. $L$ -functions of modular forms and higher regulators (book in preparation)

Scholl, A. J. Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences, Invent. math., Volume 79 (1985), pp. 49-77 | DOI | MR | Zbl

Scholl, A. J. Motives for modular forms, Invent. math., Volume 100 (1990), pp. 419-430 | DOI | MR | Zbl

Stevens, Glenn Arithmetic on modular curves, Progress in Math., 20, Birkhäuser, 1982, 214 pages | MR | Zbl | DOI

Stevens, Glenn The cuspidal group and special values of $L$ -functions, Trans. Amer. Math. Soc., Volume 291 (1985), pp. 519-550 | DOI | MR | Zbl

Skinner, Christopher; Urban, Eric The Iwasawa main conjectures for ${GL}_{2}$ , Invent. math., Volume 195 (2014), pp. 1-277 | DOI | MR | Zbl

Tsuji, Takeshi On $p$ -adic nearby cycles of log smooth families, Bull. Soc. Math. France, Volume 128 (2000), pp. 529-575 | Numdam | MR | Zbl | DOI

Tsuji, Takeshi Semi-stable conjecture of Fontaine-Jannsen: a survey, Astérisque, Volume 279 (2002), pp. 323-370 | MR | Zbl | Numdam

Ulmer, Douglas L. On the Fourier coefficients of modular forms, Ann. Sci. École Norm. Sup., Volume 28 (1995), pp. 129-160 | Numdam | MR | Zbl | DOI

Ulmer, Douglas L. On the Fourier coefficients of modular forms. II, Math. Ann., Volume 304 (1996), pp. 363-422 | DOI | MR | Zbl

Vatsal, Vinayak Canonical periods and congruence formulae, Duke Math. J., Volume 98 (1999), pp. 397-419 | DOI | MR | Zbl

Višik, M. M. Nonarchimedean measures associated with Dirichlet series, Mat. Sb. (N.S.), Volume 99 (1976), pp. 248-260 | MR | Zbl

Wach, Nathalie Représentations cristallines de torsion, Compos. math., Volume 108 (1997), pp. 185-240 | DOI | MR | Zbl

Washington, Lawrence C. Introduction to cyclotomic fields, Graduate Texts in Math., 83, Springer, New York, 1997, 487 pages | DOI | MR | Zbl

Wiles, A. On ordinary $λ$ -adic representations associated to modular forms, Invent. math., Volume 94 (1988), pp. 529-573 | DOI | MR | Zbl

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