On the Iwasawa μ-invariant and λ-invariant associated to tensor products of newforms
Annales de l'Institut Fourier, Volume 74 (2024) no. 2, pp. 451-502.

Fix an odd prime number p, and let ρ ¯ 1 ,,ρ ¯ t be a collection of two-dimensional ordinary Galois representations defined over a finite field 𝔽 p e . Suppose that we are given newforms f 1 ,,f t whose p-adic representations

ρ f 1 :G GL 2 (𝒦),,ρ f t :G GL 2 (𝒦)with𝒪 𝒦 /π 𝒦 𝔽 p e

satisfy ρ f 1 ρ f t modπ 𝒦 ρ ¯ 1 ρ ¯ t , and some other extra hypotheses. We shall determine the cyclotomic λ-invariant for the Selmer group attached to the product f 1 f t under the assumption that the μ-invariant is zero. If t=2 (i.e. the double product case) this allows us to deduce the Iwasawa Main Conjecture for f 1 f 2 if it is already known for a congruent pair f 1 f 2 , much as Greenberg and Vatsal did for t=1 (i.e. for elliptic cusp forms).

On fixe un nombre premier impair p, et soit ρ ¯ 1 ,,ρ ¯ t une famille de représentations galoisiennes ordinaires à deux dimensions définies sur un corps fini 𝔽 p e . On suppose données des formes modulaires primitives f 1 ,,f t dont les représentations p-adiques

ρ f 1 :G GL 2 (𝒦),,ρ f t :G GL 2 (𝒦)avec𝒪 𝒦 /π 𝒦 𝔽 p e

satisfont ρ f 1 ρ f t modπ 𝒦 ρ ¯ 1 ρ ¯ t , et quelques autres hypothèses supplémentaires. On détermine l’invariant λ cyclotomique pour le groupe de Selmer associé au produit f 1 f t sous l’hypothèse que l’invariant μ est nul. Si t=2 (c’est-à-dire dans le cas du double produit), cela nous permet de déduire la conjecture principale d’Iwasawa pour f 1 f 2 si elle est déjà connue pour une paire congruente f 1 f 2 , tout comme Greenberg et Vatsal l’ont fait pour t=1 (i.e. pour les formes modulaires paraboliques).

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DOI: 10.5802/aif.3593
Classification: 11F33, 11F80, 11G40, 11R23
Keywords: Galois representations, Iwasawa theory, $p$-adic $L$-functions.
Mot clés : Représentations galoisiennes, théorie d’Iwasawa, fonctions $L$ $p$-adiques.
Delbourgo, Daniel 1

1 Department of Mathematics University of Waikato Hamilton (New Zealand)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Delbourgo, Daniel. On the Iwasawa $\mu $-invariant and $\lambda $-invariant associated to tensor products of newforms. Annales de l'Institut Fourier, Volume 74 (2024) no. 2, pp. 451-502. doi : 10.5802/aif.3593. https://aif.centre-mersenne.org/articles/10.5802/aif.3593/

[1] Bloch, Spencer; Kato, Kazuya L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I (Progr. Math.), Volume 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333-400 | MR | Zbl

[2] Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard On the modularity of elliptic curves over : wild 3-adic exercises, J. Amer. Math. Soc., Volume 14 (2001) no. 4, pp. 843-939 | DOI | MR | Zbl

[3] Burns, David; Venjakob, Otmar On descent theory and main conjectures in non-commutative Iwasawa theory, J. Inst. Math. Jussieu, Volume 10 (2011) no. 1, pp. 59-118 | DOI | MR | Zbl

[4] Castella, Francesc; Kim, Chan-Ho; Longo, Matteo Variation of anticyclotomic Iwasawa invariants in Hida families, Algebra Number Theory, Volume 11 (2017) no. 10, pp. 2339-2368 | DOI | MR | Zbl

[5] Coates, J.; Sujatha, R. Fine Selmer groups of elliptic curves over p-adic Lie extensions, Math. Ann., Volume 331 (2005) no. 4, pp. 809-839 | DOI | MR | Zbl

[6] Darmon, Henri; Rotger, Victor Diagonal cycles and Euler systems II: The Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-functions, J. Amer. Math. Soc., Volume 30 (2017) no. 3, pp. 601-672 | DOI | MR | Zbl

[7] Delbourgo, Daniel On the Iwasawa Main Conjecture for the tensor product fg of two elliptic modular forms f and g (in preparation)

[8] Delbourgo, Daniel Variation of the analytic λ-invariant over a solvable extension, Proc. Lond. Math. Soc. (3), Volume 120 (2020) no. 6, pp. 918-960 | DOI | MR | Zbl

[9] Delbourgo, Daniel Variation of the algebraic λ-invariant over a solvable extension, Math. Proc. Cambridge Philos. Soc., Volume 170 (2021) no. 3, pp. 499-521 | DOI | MR | Zbl

[10] Delbourgo, Daniel; Gilmore, Hamish Controlling λ-invariants for the double and triple product p-adic L-functions, J. Théor. Nombres Bordeaux, Volume 33 (2021) no. 3.1, pp. 733-778 | DOI | MR | Zbl

[11] Delbourgo, Daniel; Lei, Antonio Congruences modulo p between ρ-twisted Hasse-Weil L-values, Trans. Amer. Math. Soc., Volume 370 (2018) no. 11, pp. 8047-8080 | DOI | MR | Zbl

[12] Deligne, Pierre Formes modulaires et représentations -adiques, Séminaire Bourbaki : vol. 1968/69, exposés 347-363 (Séminaire Bourbaki), Springer-Verlag, 1971 no. 11 (talk:355) | Zbl

[13] Emerton, Matthew; Pollack, Robert; Weston, Tom Variation of Iwasawa invariants in Hida families, Invent. Math., Volume 163 (2006) no. 3, pp. 523-580 | DOI | MR | Zbl

[14] Fontaine, Jean-Marc Périodes p-adiques, Exposé III, Séminaire du Bures-sur-Yvette, France, 1988 (Astérisque), Volume 223, Société Mathématique de France, 1994, pp. 113-184 | Zbl

[15] 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

[16] González-Sánchez, Jon; Klopsch, Benjamin Analytic pro-p groups of small dimensions, J. Group Theory, Volume 12 (2009) no. 5, pp. 711-734 | DOI | MR | Zbl

[17] 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 | DOI | MR | Zbl

[18] Greenberg, Ralph Iwasawa theory, projective modules, and modular representations, Mem. Amer. Math. Soc., Volume 211 (2011) no. 992, p. vi+185 | DOI | MR | Zbl

[19] Greenberg, Ralph; Vatsal, Vinayak On the Iwasawa invariants of elliptic curves, Invent. Math., Volume 142 (2000) no. 1, pp. 17-63 | DOI | MR | Zbl

[20] Hida, Haruzo Galois representations into GL 2 ( p [[X]]) attached to ordinary cusp forms, Invent. Math., Volume 85 (1986) no. 3, pp. 545-613 | DOI | MR | Zbl

[21] Hida, Haruzo On p-adic L-functions of GL(2)×GL(2) over totally real fields, Ann. Inst. Fourier (Grenoble), Volume 41 (1991) no. 2, pp. 311-391 | DOI | Numdam | MR | Zbl

[22] Hsieh, Ming-Lun Hida families and p-adic triple product L-functions (https://arxiv.org/abs/1705.02717, to appear in Amer. J. Math.)

[23] Hsieh, Ming-Lun; Yamana, Shunsuke Four variable p-adic triple product L-functions and the trivial zero conjecture (2019) (https://arxiv.org/abs/1906.10474)

[24] Kato, Kazuya p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applications arithmétiques. III (Astérisque), Société Mathématique de France, 2004 no. 295, pp. ix, 117-290 | Numdam | MR | Zbl

[25] Khare, Chandrashekhar; Wintenberger, Jean-Pierre Serre’s modularity conjecture. I, Invent. Math., Volume 178 (2009) no. 3, pp. 485-504 | DOI | MR | Zbl

[26] Kida, Yûji l-extensions of CM-fields and cyclotomic invariants, J. Number Theory, Volume 12 (1980) no. 4, pp. 519-528 | DOI | MR | Zbl

[27] Kings, Guido; Loeffler, David; Zerbes, Sarah Livia Rankin–Eisenstein classes and explicit reciprocity laws, Camb. J. Math., Volume 5 (2017) no. 1, pp. 1-122 | DOI | MR | Zbl

[28] Lim, Meng Fai Comparing the π-primary submodules of the dual Selmer groups, Asian J. Math., Volume 21 (2017) no. 6, pp. 1153-1181 | DOI | MR | Zbl

[29] Lim, Meng Fai 𝔐 H (G)-property and congruence of Galois representations, J. Ramanujan Math. Soc., Volume 33 (2018) no. 1, pp. 37-74 | MR | Zbl

[30] Livné, Ron On the conductors of mod l Galois representations coming from modular forms, J. Number Theory, Volume 31 (1989) no. 2, pp. 133-141 | DOI | MR | Zbl

[31] Mazur, B.; Tate, J.; Teitelbaum, J. On p-adic analogues of the conjectures of Birch and Swinnerton–Dyer, Invent. Math., Volume 84 (1986) no. 1, pp. 1-48 | DOI | MR | Zbl

[32] Nekovář, Jan On p-adic height pairings, Séminaire de Théorie des Nombres, Paris, 1990–91 (Progr. Math.), Volume 108, Birkhäuser Boston, Boston, MA, 1993, pp. 127-202 | DOI | MR | Zbl

[33] Panchishkin, Alexey A. Non-Archimedean L-functions of Siegel and Hilbert modular forms, Lecture Notes in Mathematics, 1471, Springer-Verlag, Berlin, 1991, vi+157 pages | DOI | MR | Zbl

[34] Serre, Jean-Pierre Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1972) no. 4, pp. 259-331 | DOI | MR | Zbl

[35] Serre, Jean-Pierre Sur les représentations modulaires de degré 2 de Gal( ¯/), Duke Math. J., Volume 54 (1987) no. 1, pp. 179-230 | DOI | MR | Zbl

[36] Shahidi, Freydoon On certain L-functions, Amer. J. Math., Volume 103 (1981) no. 2, pp. 297-355 | DOI | MR | Zbl

[37] Shimura, Goro The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., Volume 29 (1976) no. 6, pp. 783-804 | DOI | MR | Zbl

[38] Skinner, Christopher; Urban, Eric The Iwasawa main conjectures for GL 2 , Invent. Math., Volume 195 (2014) no. 1, pp. 1-277 | DOI | MR | Zbl

[39] Weston, Tom Iwasawa invariants of Galois deformations, Manuscripta Math., Volume 118 (2005) no. 2, pp. 161-180 | DOI | MR | Zbl

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