Skip to main content
SpringerLink
Log in

On the anticyclotomic Iwasawa main conjecture for Hilbert modular forms of parallel weights

  • Published:
Annales mathématiques du Québec Aims and scope Submit manuscript

Abstract

In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove an one-sided divisibility result toward the Iwasawa main conjecture in this setting. The proof relies on the first and second reciprocity laws relating theta elements to Heegner point Euler systems on Shimura curves. As a by-product we also prove a result towards the rank 0 case of certain Bloch–Kato conjecture and a parity conjecture.

Résumé

Dans cet article, nous étudions la théorie d’Iwasawa pour les formes modulaires de Hilbert sur l’extension anticyclotomique d’un champ CM. Nous prouvons un résultat de divisibilité unilatérale vers la conjecture principale d’Iwasawa dans ce cadre. La preuve repose sur les première et deuxième lois de réciprocité reliant les éléments theta aux systèmes d’Euler de points de Heegner sur les courbes de Shimura. En tant que sous-produit, nous prouvons également un résultat vers le cas de rang 0 d’une certaine conjecture de Bloch–Kato et d’une conjecture de parité.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Aflalo and J. Nekovář. Non-triviality of CM points in ring class field towers. Israel J. Math., 175:225–284, 2010. With an appendix by Christophe Cornut.

  2. M. Bertolini and H. Darmon. Heegner points on Mumford-Tate curves. Invent. Math., 126(3):413–456, 1996.

  3. M. Bertolini and H. Darmon. Iwasawa’s main conjecture for elliptic curves over anticyclotomic \({{\mathbb{Z}}_{p}}\)-extensions. Ann. of Math. (2), 162(1):1–64, 2005.

  4. S. Bloch and K. Kato. \(L\)-functions and Tamagawa numbers of motives. In The Grothendieck Festschrift, Vol. I, volume 86 of Progr. Math., pages 333–400. Birkhäuser Boston, Boston, MA, 1990.

  5. S. Bosch, W. Lütkebohmert, and M. Raynaud. Néron models, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1990.

  6. N. Boston, H. Lenstra, Jr., and Kenneth A. Ribet. Quotients of group rings arising from two-dimensional representations. C. R. Acad. Sci. Paris Sér. I Math., 312(4):323–328, 1991.

  7. D. Blasius and J. Rogawski. Motives for Hilbert modular forms. Invent. Math., 114(1):55–87, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Carayol. Sur les représentations \(l\)-adiques associées aux formes modulaires de Hilbert. Ann. Sci. École Norm. Sup. (4), 19(3):409–468, 1986.

  9. H. Carayol. Sur la mauvaise réduction des courbes de Shimura. Compositio Math., 59(2):151–230, 1986.

  10. F. Castella and M.-L. Hsieh. Heegner cycles and \(p\)-adic L-functions. Math. Ann, 370:567–628, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  11. I. V. Čerednik. Uniformization of algebraic curves by discrete arithmetic subgroups of \({\rm PGL}_{2}(k_{w})\) with compact quotient spaces. Mat. Sb. (N.S.), 100(142)(1):59–88, 165, 1976.

  12. C. Cheng. Ihara’s lemma for shimura curves over totally real fields and multiplicity two. Preprint, 2011.

  13. M. Chida and M. Hsieh. On the anticyclotomic Iwasawa main conjecture for modular forms. Compos. Math., 151(5):863–897, 2015.

  14. M. Chida and M. Hsieh. Special values of anticyclotomic \(L\)-functions for modular forms. J. Reine Angew. Math., 741:87–131, 2018.

  15. C. Cornut and V. Vatsal. Nontriviality of Rankin-Selberg \(L\)-functions and CM points. 320:121–186, 2007.

  16. F. Diamond. An extension of Wiles’ results. In Modular forms and Fermat’s last theorem (Boston, MA, 1995), pages 475–489. Springer, New York, 1997.

  17. F. Diamond. The Taylor-Wiles construction and multiplicity one. Invent. Math., 128(2):379–391, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Diamond and R. Taylor. Nonoptimal levels of mod \(l\) modular representations. Invent. Math., 115(3):435–462, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Greenberg. Iwasawa theory for elliptic curves. In Arithmetic theory of elliptic curves (Cetraro, 1997), volume 1716 of Lecture Notes in Math., pages 51–144. Springer, Berlin, 1999.

  20. R. Greenberg and V. Vatsal. On the Iwasawa invariants of elliptic curves. Invent. Math., 142(1):17–63, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Hida. On p-adic hecke algebras for GL(2) over totally real fields. Annals of Mathematics, 128:295–384, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Howard. Bipartite Euler systems. J. Reine Angew. Math., 597:1–25, 2006.

  23. P. Hung. On the non-vanishing \({\rm mod}\, \ell \) of central \(L\)-values with anticyclotomic twists for Hilbert modular forms. J. Number Theory, 173:170–209, 2017.

  24. Y. Ihara. Shimura curves over finite fields and their rational points. In Applications of curves over finite fields (Seattle, WA, 1997), volume 245 of Contemp. Math., pages 15–23. Amer. Math. Soc., Providence, RI, 1999.

  25. F. Jarvis. Level lowering for modular mod \(l\) representations over totally real fields. Math. Ann., 313(1):141–160, 1999.

  26. B. Jordan and R. Livné. Local Diophantine properties of Shimura curves. Math. Ann., 270(2):235–248, 1985.

  27. C. Kim, R. Pollack, and T. Weston. On the freeness of anticyclotomic Selmer groups of modular forms. Int. J. Number Theory, 13(6):1443–1455, 2017.

  28. M. Longo. Anticyclotomic Iwasawa’s main conjecture for Hilbert modular forms. Comment. Math. Helv., 87(2):303–353, 2012.

  29. J. Manning and J. Shotton. Ihara’s lemma for shimura curves over totally real fields via patching. Math. Ann., 379:187–234, 2021.

    Article  MathSciNet  MATH  Google Scholar 

  30. J. S. Milne. Arithmetic duality theorems. BookSurge, LLC, Charleston, SC, second edition, 2006.

    MATH  Google Scholar 

  31. J. Nekovář. Selmer complexes. Astérisque, (310):viii+559, 2006.

  32. J.  Nekovář. On the parity of ranks of Selmer groups. III. Doc. Math., 12:243–274, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Nekovář. Growth of Selmer groups of Hilbert modular forms over ring class fields. Ann. Sci. Éc. Norm. Supér. (4), 41(6):1003–1022, 2008.

  34. J. Nekovář. Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two. Canad. J. Math., 64(3):588–668, 2012.

  35. R. Pollack and T. Weston. On anticyclotomic \(\mu \)-invariants of modular forms. Compos. Math., 147(5):1353–1381, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Rajaei. On the levels of mod \(l\) Hilbert modular forms. J. Reine Angew. Math., 537:33–65, 2001.

  37. K. Ribet. On modular representations of \({\rm Gal}(\overline{\bf Q}/{\bf Q})\) arising from modular forms. Invent. Math., 100(2):431–476, 1990.

  38. K. Rubin. Euler systems, volume 147 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2000. Hermann Weyl Lectures. The Institute for Advanced Study.

  39. G. Shimura. The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J., 45(3):637–679, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  40. R. Taylor. On Galois representations associated to Hilbert modular forms. Invent. Math., 98(2):265–280, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  41. R. Taylor. On the meromorphic continuation of degree two \(L\)-functions. Doc. Math., (Extra Vol.):729–779 (electronic), 2006.

  42. J. Van Order. On the dihedral main conjectures of Iwasawa theory for Hilbert modular eigenforms. Canad. J. Math., 65(2):403–466, 2013.

  43. V. Vatsal. Uniform distribution of Heegner points. Invent. Math., 148(1):1–46, 2002.

  44. V. Vatsal. Special values of anticyclotomic \(L\)-functions. Duke Math. J., 116(2):219–261, 2003.

  45. X. Wan. The Iwasawa main conjecture for Hilbert modular forms. Forum Math. Sigma, 3:Paper No. e18, 95, 2015.

  46. A. Wiles. On ordinary \(\lambda \)-adic representations associated to modular forms. Invent. Math., 94(3):529–573, 1988.

  47. S. Zhang. Heights of Heegner points on Shimura curves. Ann. of Math. (2), 153(1):27–147, 2001.

  48. W. Zhang. Selmer groups and the indivisibility of Heegner points. Camb. J. Math., 2(2):191–253, 2014.

Download references

Acknowledgements

This article is essentially author’s Ph.D thesis at Pennsylvania State university. I would like to thank my advisor Professor Wen-Ching Winnie Li for constant help and encouragement during my graduate studies. I would like to express my gratitude to my thesis advisor Professor Ming-Lun Hsieh for generously sharing his knowledge with me and answering numerous questions not restricted to the topic of this thesis. I would like to thank the referees for all the comments and helpful suggestions.

Author information

Authors and Affiliations

  1. Shanghai Center for Mathematical Sciences, Fudan University, No. 2005 Songhu Road, Shanghai, 200438, China

    Haining Wang

Authors
  1. Haining Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haining Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, H. On the anticyclotomic Iwasawa main conjecture for Hilbert modular forms of parallel weights. Ann. Math. Québec 47, 195–248 (2023). https://doi.org/10.1007/s40316-022-00208-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40316-022-00208-7

Keywords

Mathematics Subject Classification

Search

Navigation