Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes
Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 673-707.

Let f be a cuspidal newform with complex multiplication (CM) and let p be an odd prime at which f is non-ordinary. We construct admissible p-adic L-functions for the symmetric powers of f, thus verifying conjectures of Dabrowski and Panchishkin in this special case. We combine this with recent work of Benois to prove the trivial zero conjecture in this setting. We also construct “mixed” plus and minus p-adic L-functions and prove an analogue of Pollack’s decomposition of the admissible p-adic L-functions. On the arithmetic side, we define corresponding mixed plus and minus Selmer groups and formulate the Main Conjecture of Iwasawa Theory.

Soit f une forme primitive cuspidale à multiplication complexe (CM) et soit p un nombre premier impair tel que f soit non-ordinaire en p. Nous construisons des fonctions L p-adiques admissibles pour les puissances symétriques de f, vérifiant ainsi des cas particuliers de conjectures de Dabrowski et Panchishkin. À l’aide d’un résultat récent de Benois, nous prouvons la conjecture des zéros triviaux dans notre contexte. De plus, nous construisons des fonctions L p-adiques plus/moins “mixtes” et obtenons une décomposition des fonctions L p-adiques admissibles analogue à celle de Pollack. Du côté arithmétique, nous définissons les groupes de Selmer plus/moins mixtes correspondants et nous énonçons une Conjecture Principale.

DOI: 10.5802/jtnb.885
Classification: 11R23, 11F80, 11F67
Robert Harron 1; Antonio Lei 2

1 Department of Mathematics Keller Hall University of Hawai‘i at Mānoa Honolulu, HI 96822, USA
2 Département de mathématiques et de statistique Pavillon Alexandre-Vachon Université Laval Québec, QC Canada G1V 0A6
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Robert Harron; Antonio Lei. Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 673-707. doi : 10.5802/jtnb.885. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.885/

[AV75] Y. Amice and J. Vélu, Distributions p-adiques associées aux séries de Hecke, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), Soc. Math. France, Paris, (1975), 119–131. Astérisque, Nos. 24–25. | MR | Zbl

[BC79] A. Borel and W. Casselman (eds.), Automorphic forms, representations, and L-functions, Proceedings of the Symposium in Pure Mathematics, vol. 33, American Mathematical Society, (1979), In two parts. | Zbl

[Ben11] D. Benois, A generalization of Greenberg’s -invariant, Amer. J. Math. 133, (2011), 6, 1573–1632. | MR

[Ben13] D. Benois, Trivial zeros of p-adic L-functions at near central points, online at J. Inst. Math. Jussieu, doi: http://dx.doi.org/10.1017/S1474748013000261 | MR

[CGMzS89] J. Coates, R. Greenberg, B. Mazur, and I. Satake (eds.), Algebraic number theory, Advanced Studies in Pure Mathematics, vol. 17, Academic Press, (1989), Papers in honor of Kenkichi Iwasawa on the occasion of his 70th birthday on September 11, 1987. | MR | Zbl

[CloMi90] L. Clozel and J. S. Milne (eds.), Automorphic forms, Shimura varieties, and L-functions, Vol. II, Perspectives in Mathematics, vol. 11, Academic Press, 1990, Proceedings of the conference held at the University of Michigan, Ann Arbor, Michigan, July 6–16, 1988. | MR | Zbl

[Col98] P. Colmez, Théorie d’Iwasawa des représentations de de Rham d’un corps local, Ann. of Math. (2) 148, (1998), 2, 485–571. | MR | Zbl

[CSch87] J. Coates and C.-G. Schmidt, Iwasawa theory for the symmetric square of an elliptic curve, J. Reine Angew. Math. 375/376, (1987), 104–156. | EuDML | MR | Zbl

[Dab93] A. Dabrowski, Admissible p-adic L-functions of automorphic forms, Moscow Univ. Math. Bull. 48, (1993), 2, 6–10, English translation of original Russian. | MR | Zbl

[Dab11] A. Dabrowski, Bounded p-adic L-functions of motives at supersingular primes, C. R. Math. Acad. Sci. Paris 349, (2011), 7–8, 365–368. | MR | Zbl

[D69] P. Deligne, Formes modulaires et représentations l-adiques, Séminaire Bourbaki (1968/69), no. 21, Exp. No. 355, 139–172. | Numdam | Zbl

[D79] P. Deligne, Valeurs de fonctions L et périodes d’intégrales, [BC79], part 2, (1979), 313–346. | MR | Zbl

[G89] R. Greenberg, Iwasawa theory for p-adic representations, [CGMzS89], (1989), 97–137. | MR | Zbl

[G94] R. Greenberg, Trivial zeroes of p-adic L-functions, [MzSt94], (1994), 149–174. | MR | Zbl

[H13] R. Harron, The exceptional zero conjecture for symmetric powers of CM modular forms: the ordinary case, Int. Math. Res. Not. 2013, (2013), 16, art. ID rns161, 3744–3770. | MR

[Hi88] H. Hida, Modules of congruence of Hecke algebras and L-functions associated with cusp forms, Amer. J. Math. 110, (1988), 2, 323–382. | MR | Zbl

[Hi90] H. Hida, p-adic L-functions for base change lifts of GL 2 to GL 3 , [CloMi90], (1990), 93–142. | MR | Zbl

[Iw72] K. Iwasawa, Lectures on p-adic L-functions, Annals of Mathematics Studies, vol. 74, Princeton University Press, (1972). | MR | Zbl

[JS76] H. Jacquet and J. A. Shalika, A non-vanishing theorem for zeta functions of GL n , Invent. Math. 38, (1976), 1, 1–16. | MR | Zbl

[KL64] T. Kubota and H.-W. Leopoldt, Eine p-adische Theorie der Zetawerte. I. Einführung der p-adischen Dirichletschen L-Funktionen, J. Reine Angew. Math. 214/215, (1964), 328–339. | MR | Zbl

[KPZ10] B. D. Kim, J. Park, and B. Zhang, Iwasawa main conjecture for CM elliptic curves over abelian extensions at supersingular primes, preprint, (2010).

[L11] A. Lei, Iwasawa theory for modular forms at supersingular primes, Compos. Math. 147, (2011), 03, 803–838. | MR | Zbl

[L12] A. Lei, Iwasawa theory for the symmetric square of a CM modular form at inert primes, Glasg. Math. J. 54, (2012), 02, 241–259. | MR | Zbl

[LLZ11] A. Lei, D. Loeffler, and S. L. Zerbes, Coleman maps and the p-adic regulator, Algebra & Number Theory 5, (2011), 8, 1095–1131. | MR | Zbl

[MzSt94] B. Mazur and G. Stevens (eds.), p-adic monodromy and the Birch and Swinnerton-Dyer conjecture, Contemporary Mathematics, vol. 165, American Mathematical Society, 1994, Papers from the workshop held at Boston University, (1991), August 12–16. | MR | Zbl

[MzW84] B. Mazur and A. Wiles, Class fields of abelian extensions of , Invent. Math., textbf76, (1984), 2, 179–330. | MR | Zbl

[Och00] T. Ochiai, Control theorem for Bloch–Kato’s Selmer groups of p-adic representations, J. Number Theory 82, (2000), 1, 69–90. | MR | Zbl

[Pan94] A. Panchishkin, Motives over totally real fields and p-adic L-functions, Ann. Inst. Fourier (Grenoble) 44, (1994), 4, 989–1023. | Numdam | MR | Zbl

[Pol03] R. Pollack, On the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118, (2003), 3, 523–558. | MR | Zbl

[PolRu04] R. Pollack and Karl Rubin, The main conjecture for CM elliptic curves at supersingular primes, Ann. of Math. (2) 159 (2004), 1, 447–464. | MR | Zbl

[PR94] B. Perrin-Riou, Théorie d’Iwasawa des représentations p-adiques sur un corps local, Invent. Math. 115 (1994), 1, 81–161. | MR | Zbl

[Rib77] K. A. Ribet, Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Springer, Berlin, 1977, 17–51. Lecture Notes in Math., Vol. 601. | MR | Zbl

[Sai97] T. Saito, Modular forms and p-adic Hodge theory, Invent. Math. 129 (1997), 3, 607–620. | MR | Zbl

[Sch88] C.-G. Schmidt, p-adic measures attached to automorphic representations of GL(3), Invent. Math. 92 (1988), 3, 597–631. | MR | Zbl

[U06] E. Urban, Groupes de Selmer et fonctions L p-adiques pour les représentations modulaires adjointes. Available at http://www.math.columbia.edu/~urban/EURP.html, 2006.

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