Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 13, 2017

Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms

  • Kâzım Büyükboduk and Antonio Lei ORCID logo EMAIL logo
From the journal Forum Mathematicum

Abstract

This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic p-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson–Flach elements.

MSC 2010: 11R23; 11F11; 11R20

Communicated by Jan Bruinier


Funding source: European Commission

Award Identifier / Grant number: 745691

Award Identifier / Grant number: 05710

Funding statement: The first author is partially supported by the European Commission Global Fellowship CriticalGZ. The second author is supported by the NSERC Discovery Grants Program 05710.

Acknowledgements

We would like to thank Henri Darmon and Chan-Ho Kim for enlightening discussions during the preparation of this paper, and the anonymous referees for the suggestions and comments on a previous version of this paper.

References

[1] A. Agboola and B. Howard, Anticyclotomic Iwasawa theory of CM elliptic curves. II, Math. Res. Lett. 12 (2005), no. 5–6, 611–621. 10.4310/MRL.2005.v12.n5.a1Search in Google Scholar

[2] A. Agboola and B. Howard, Anticyclotomic Iwasawa theory of CM elliptic curves, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1001–1048. 10.5802/aif.2206Search in Google Scholar

[3] T. Arnold, Anticyclotomic main conjectures for CM modular forms, J. Reine Angew. Math. 606 (2007), 41–78. 10.1515/CRELLE.2007.034Search in Google Scholar

[4] M. Bertolini and H. Darmon, Iwasawa’s main conjecture for elliptic curves over anticyclotomic p-extensions, Ann. of Math. (2) 162 (2005), no. 1, 1–64. 10.4007/annals.2005.162.1Search in Google Scholar

[5] M. Bertolini, H. Darmon and K. Prasanna, Generalized Heegner cycles and p-adic Rankin L-series, Duke Math. J. 162 (2013), no. 6, 1033–1148. 10.1215/00127094-2142056Search in Google Scholar

[6] K. Büyükboduk, Main conjectures for CM fields and a Yager-type theorem for Rubin–Stark elements, Int. Math. Res. Not. IMRN 2014 (2014), no. 21, 5832–5873. 10.1093/imrn/rnt140Search in Google Scholar

[7] K. Büyükboduk, On the anticyclotomic Iwasawa theory of CM forms at supersingular primes, Rev. Mat. Iberoam. 31 (2015), no. 1, 109–126. 10.4171/RMI/828Search in Google Scholar

[8] K. Büyükboduk, Deformations of Kolyvagin systems, Ann. Math. Qué. 40 (2016), no. 2, 251–302. 10.1007/s40316-015-0044-4Search in Google Scholar

[9] K. Büyükboduk and A. Lei, Coleman-adapted Rubin–Stark Kolyvagin systems and supersingular Iwasawa theory of CM abelian varieties, Proc. Lond. Math. Soc. (3) 111 (2015), no. 6, 1338–1378. 10.1112/plms/pdv054Search in Google Scholar

[10] K. Büyükboduk and A. Lei, Iwasawa theory of elliptic modular forms over imaginary quadratic fields at non-ordinary primes, preprint (2016), https://arxiv.org/abs/1605.05310. 10.1093/imrn/rnz117Search in Google Scholar

[11] K. Büyükboduk and T. Ochiai, Main conjectures for higher rank nearly ordinary families. I, preprint (2017), https://arxiv.org/abs/1708.04494. Search in Google Scholar

[12] F. Castella, p-adic heights of Heegner points and Beilinson–Flach classes, J. Lond. Math. Soc. (2) 96 (2017), no. 1, 156–180. 10.1112/jlms.12058Search in Google Scholar

[13] F. Castella, C.-H. Kim and M. Longo, Variation of anticyclotomic Iwasawa invariants in Hida families, preprint (2015), https://arxiv.org/abs/1504.06310; to appear in Algebra Number Theory. 10.2140/ant.2017.11.2339Search in Google Scholar

[14] M. Chida and M.-L. Hsieh, On the anticyclotomic Iwasawa main conjecture for modular forms, Compos. Math. 151 (2015), no. 5, 863–897. 10.1112/S0010437X14007787Search in Google Scholar

[15] M. Chida and M.-L. Hsieh, Special values of anticyclotomic L-functions for modular forms, J. Reine Angew. Math. (2016), 10.1515/crelle-2015-0072. 10.1515/crelle-2015-0072Search in Google Scholar

[16] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton–Dyer, Invent. Math. 39 (1977), no. 3, 223–251. 10.1007/BF01402975Search in Google Scholar

[17] C. Cornut and V. Vatsal, Nontriviality of Rankin–Selberg L-functions and CM points, L-Functions and Galois Representations, London Math. Soc. Lecture Note Ser. 320, Cambridge University Press, Cambridge (2007), 121–186. 10.1017/CBO9780511721267.005Search in Google Scholar

[18] H. Darmon and A. Iovita, The anticyclotomic main conjecture for elliptic curves at supersingular primes, J. Inst. Math. Jussieu 7 (2008), no. 2, 291–325. 10.1017/S1474748008000042Search in Google Scholar

[19] M. Emerton, R. Pollack and T. Weston, Variation of Iwasawa invariants in Hida families, Invent. Math. 163 (2006), no. 3, 523–580. 10.1007/s00222-005-0467-7Search in Google Scholar

[20] H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms. I, Invent. Math. 79 (1985), no. 1, 159–195. 10.1007/BF01388661Search in Google Scholar

[21] H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms. II, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 3, 1–83. 10.5802/aif.1141Search in Google Scholar

[22] H. Hida, Elementary Theory of L-Functions and Eisenstein Series, London Math. Soc. Student Texts 26, Cambridge University Press, Cambridge, 1993. 10.1017/CBO9780511623691Search in Google Scholar

[23] H. Hida and J. Tilouine, Anti-cyclotomic Katz p-adic L-functions and congruence modules, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 2, 189–259. 10.24033/asens.1671Search in Google Scholar

[24] B. Howard, The Heegner point Kolyvagin system, Compos. Math. 140 (2004), no. 6, 1439–1472. 10.1112/S0010437X04000569Search in Google Scholar

[25] B. Howard, The Iwasawa theoretic Gross–Zagier theorem, Compos. Math. 141 (2005), no. 4, 811–846. 10.1112/S0010437X0500134XSearch in Google Scholar

[26] M.-L. Hsieh, Special values of anticyclotomic Rankin–Selberg L-functions, Doc. Math. 19 (2014), 709–767. 10.4171/dm/462Search in Google Scholar

[27] H. Jacquet, Automorphic Forms on GL(2). Part II, Lecture Notes in Math. 278, Springer, Berlin, 1972. 10.1007/BFb0058503Search in Google Scholar

[28] H. Jacquet and R. P. Langlands, Automorphic Forms on GL(2), Lecture Notes in Math. 114, Springer, Berlin, 1970. 10.1007/BFb0058988Search in Google Scholar

[29] K. Kato, Generalized explicit reciprocity laws, Adv. Stud. Contemp. Math. (Pusan) 1 (1999), 57–126. Search in Google Scholar

[30] K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-Adiques et Applications Arithmétiques. III, Astérisque 295, Société Mathématique de France, Paris (2004), 117–290. Search in Google Scholar

[31] G. Kings, D. Loeffler and S. Zerbes, Rankin–Selberg Euler systems and p-adic interpolation, preprint (2014), https://arxiv.org/abs/1405.3079. Search in Google Scholar

[32] G. Kings, D. Loeffler and S. L. Zerbes, Rankin–Eisenstein classes for modular forms, preprint (2015), https://arxiv.org/abs/1501.03289. 10.1353/ajm.2020.0002Search in Google Scholar

[33] G. Kings, D. Loeffler and S. L. Zerbes, Rankin–Eisenstein classes and explicit reciprocity laws, Camb. J. Math. 5 (2017), no. 1, 1–122. 10.4310/CJM.2017.v5.n1.a1Search in Google Scholar

[34] A. Lei, D. Loeffler and S. L. Zerbes, Euler systems for Rankin–Selberg convolutions of modular forms, Ann. of Math. (2) 180 (2014), no. 2, 653–771. 10.4007/annals.2014.180.2.6Search in Google Scholar

[35] A. Lei, D. Loeffler and S. L. Zerbes, Euler systems for modular forms over imaginary quadratic fields, Compos. Math. 151 (2015), no. 9, 1585–1625. 10.1112/S0010437X14008033Search in Google Scholar

[36] D. Loeffler, O. Venjakob and S. L. Zerbes, Local epsilon isomorphisms, Kyoto J. Math. 55 (2015), no. 1, 63–127. 10.1215/21562261-2848124Search in Google Scholar

[37] B. Mazur and K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (2004), no. 799, 1–96. 10.1090/memo/0799Search in Google Scholar

[38] J. Nekovář, On p-adic height pairings, Séminaire de Théorie des Nombres (Paris 1990–91), Progr. Math. 108, Birkhäuser, Boston (1993), 127–202. 10.1007/978-1-4757-4271-8_8Search in Google Scholar

[39] J. Nekovář, Selmer Complexes, Astérisque 310, Société Mathématique de France, Paris, 2007. Search in Google Scholar

[40] M. Ohta, Ordinary p-adic étale cohomology groups attached to towers of elliptic modular curves, Compos. Math. 115 (1999), no. 3, 241–301. 10.1023/A:1000556212097Search in Google Scholar

[41] B. Perrin-Riou, Arithmétique des courbes elliptiques et théorie d’Iwasawa, Mém. Soc. Math. France (N.S.) (1984), no. 17, 1–130. 10.24033/msmf.318Search in Google Scholar

[42] B. Perrin-Riou, Fonctions Lp-adiques associées à une forme modulaire et à un corps quadratique imaginaire, J. Lond. Math. Soc. (2) 38 (1988), no. 1, 1–32. 10.1112/jlms/s2-38.1.1Search in Google Scholar

[43] R. Pollack and K. Rubin, The main conjecture for CM elliptic curves at supersingular primes, Ann. of Math. (2) 159 (2004), no. 1, 447–464. 10.4007/annals.2004.159.447Search in Google Scholar

[44] D. E. Rohrlich, L-functions and division towers, Math. Ann. 281 (1988), no. 4, 611–632. 10.1007/BF01456842Search in Google Scholar

[45] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25–68. 10.1007/BF01239508Search in Google Scholar

[46] C. Skinner and E. Urban, The Iwasawa main conjectures for GL2, Invent. Math. 195 (2014), no. 1, 1–277. 10.1007/s00222-013-0448-1Search in Google Scholar

[47] X. Wan, Iwasawa main conjecture for Rankin–Selberg p-adic L-functions, preprint (2014), http://www.mcm.ac.cn/faculty/wx/201609/W020161128403554113041.pdf. 10.2140/ant.2020.14.383Search in Google Scholar

[48] X. Wan, The Iwasawa main conjecture for Hilbert modular forms, Forum Math. Sigma 3 (2015), Article ID e18. 10.1017/fms.2015.16Search in Google Scholar

[49] A. Wiles, Higher explicit reciprocity laws, Ann. Math. (2) 107 (1978), no. 2, 235–254. 10.2307/1971143Search in Google Scholar

Received: 2016-02-26
Revised: 2017-10-31
Published Online: 2017-12-13
Published in Print: 2018-07-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/forum-2016-0189/html
Scroll to top button