Skip to main content
SpringerLink
Log in

\(\Lambda \)-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Let p be an odd prime and K an imaginary quadratic field where p splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a p-ordinary elliptic curve over the anticyclotomic \({\mathbb {Z}}_p\)-extension of K does not admit any proper \(\Lambda \)-submodule of finite index, where \(\Lambda \) is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of p-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have \(\Lambda \)-corank one, so they are not \(\Lambda \)-cotorsion. As an application of our main theorem, we prove results in the vein of Greenberg–Vatsal on Iwasawa invariants of p-congruent elliptic curves, extending to the supersingular case results for p-ordinary elliptic curves due to Hatley–Lei.

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. Bertolini, M.: An annihilator for the \(p\)-Selmer group by means of Heegner points, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 5(2), 129–140 (1994)

  2. Bertolini, M.: Selmer groups and Heegner points in anticyclotomic \( { Z}_p\)-extensions. Compos. Math. 99(2), 153–182 (1995)

    MATH  Google Scholar 

  3. Bertolini, M.: Iwasawa theory for elliptic curves over imaginary quadratic fields. J. Théor. Nombres Bordeaux 13(1), 1–25 (2001)

    Article  MathSciNet  Google Scholar 

  4. Bertolini, M., Darmon, H.: Derived heights and generalized Mazur-Tate regulators. Duke Math. J. 76(1), 75–111 (1994)

    Article  MathSciNet  Google Scholar 

  5. Burungale, A.A.: On the non-triviality of the \(p\)-adic Abel-Jacobi image of generalised Heegner cycles modulo \(p\), I: modular curves. J. Algebraic Geom. 29(2), 329–371 (2020)

    Article  MathSciNet  Google Scholar 

  6. Castella, F., Kim, C.-H., Longo, M.: Variation of anticyclotomic Iwasawa invariants in Hida families. Algebra Number Theory 11(10), 2339–2368 (2017)

    Article  MathSciNet  Google Scholar 

  7. Castella, F., Wan, X.: Perrin–Riou’s main conjecture for elliptic curves at supersingular primes. arXiv:1607.02019 (2016)

  8. Çiperiani, M.: Tate-Shafarevich groups in anticyclotomic \({\mathbb{Z}}_p\)-extensions at supersingular primes. Compos. Math. 145(2), 293–308 (2009)

    Article  MathSciNet  Google Scholar 

  9. Darmon, H., Iovita, A.: The anticyclotomic Main Conjecture for elliptic curves at supersingular primes. J. Inst. Math. Jussieu 7(2), 291–325 (2008)

    Article  MathSciNet  Google Scholar 

  10. Emerton, M., Pollack, R., Weston, T.: Variation of Iwasawa invariants in Hida families. Invent. Math. 163(3), 523–580 (2006)

    Article  MathSciNet  Google Scholar 

  11. Greenberg, R., Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, : Lecture Notes in Math., vol. 1716. Springer, Berlin 1999, 51–144 (1997)

  12. Greenberg, R.: Surjectivity of the global-to-local map defining a Selmer group. Kyoto J. Math. 50(4), 853–888 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Greenberg, R.: On the structure of Selmer groups, Elliptic curves, modular forms and Iwasawa theory, Springer Proc. Math. Stat., vol. 188, Springer, Cham, pp. 225–252 (2016)

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

    Article  MathSciNet  Google Scholar 

  15. Hatley, J., Lei, A.: Arithmetic properties of signed Selmer groups at non-ordinary primes. Ann. Inst. Fourier (Grenoble) 69(3), 1259–1294 (2019)

    Article  MathSciNet  Google Scholar 

  16. Hatley, J., Lei, A.: Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms. Math. Res. Lett. 26(4), 1115–1144 (2019)

    Article  MathSciNet  Google Scholar 

  17. Hatley, J., Lei, A.: Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms – Part II, arXiv:2009.03772 (2020)

  18. Iovita, A., Pollack, R.: Iwasawa theory of elliptic curves at supersingular primes over \({\mathbb{Z}}_p\)-extensions of number fields. J. Reine Angew. Math. 598, 71–103 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Jannsen, U.: Iwasawa modules up to isomorphism, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, pp. 171–207 (1989)

  20. Kidwell, K.: On the structure of Selmer groups of \(p\)-ordinary modular forms over \({\mathbf{Z}}_p\)-extensions. J. Number Theory 187, 296–331 (2018)

    Article  MathSciNet  Google Scholar 

  21. Kim, B.D.: The parity conjecture for elliptic curves at supersingular reduction primes. Compos. Math. 143(1), 47–72 (2007)

    Article  MathSciNet  Google Scholar 

  22. Kim, B.D.: The Iwasawa invariants of the plus/minus Selmer groups. Asian J. Math. 13(2), 181–190 (2009)

    Article  MathSciNet  Google Scholar 

  23. Kim, B.D.: Signed-Selmer groups over the \(\mathbb{Z}_p^2\)-extension of an imaginary quadratic field. Canad. J. Math. 66(4), 826–843 (2014)

    Article  MathSciNet  Google Scholar 

  24. Kitajima, T., Otsuki, R.: On the plus and the minus Selmer groups for elliptic curves at supersingular primes. Tokyo J. Math. 41(1), 273–303 (2018)

    Article  MathSciNet  Google Scholar 

  25. Kobayashi, S.: Iwasawa theory for elliptic curves at supersingular primes. Invent. Math. 152(1), 1–36 (2003)

    Article  MathSciNet  Google Scholar 

  26. Kolyvagin, V.A.: Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, pp. 435–483 (1990)

  27. Longo, M., Vigni, S.: Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes. Boll. Unione Mat. Ital. 12(3), 315–347 (2019)

    Article  MathSciNet  Google Scholar 

  28. Matar, A.: Plus/minus Selmer groups and anticyclotomic \({\mathbf{Z}}_p\)-extensions, available at http://www.ahmedmatar.net/Anticyclotomic_paper4_v2.pdf (2019)

  29. Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18(3–4), 183–266 (1972)

    Article  MathSciNet  Google Scholar 

  30. Milne, J.: Arithmetic Duality Theorems, Perspectives in Mathematics, vol. 1. Academic Press Inc, Boston, MA (1986)

    Google Scholar 

  31. Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323. Springer, Berlin (2000)

    MATH  Google Scholar 

  32. Perrin-Riou, B.: Fonctions \(L\)\(p\)-adiques, théorie d’Iwasawa et points de Heegner. Bull. Soc. Math. France 115(4), 399–456 (1987)

    Article  MathSciNet  Google Scholar 

  33. Pollack, R.: On the \(p\)-adic \(L\)-function of a modular form at a supersingular prime. Duke Math. J. 118(3), 523–558 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Ponsinet, G.: On the structure of signed Selmer groups. Math. Z. 294(3–4), 1635–1658 (2020)

    Article  MathSciNet  Google Scholar 

  36. Suman, A., Lim, M.F.: On the signed Selmer groups of congruent elliptic curves with semistable reduction at all primes above \(p\) (2019)

  37. Weston, T.: Iwasawa invariants of Galois deformations. Manuscripta Math. 118(2), 161–180 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Mirela Çiperiani and Matteo Longo for interesting discussions on some of the topics of this paper. We are also grateful to the anonymous referee for several helpful suggestions which clarified some proofs and otherwise improved the exposition of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Union College, Bailey Hall 202, Schenectady, NY, 12308, USA

    Jeffrey Hatley

  2. Département de mathématiques et de statistique, Pavillon Alexandre-Vachon, Université Laval, Quebec, QC, G1V 0A6, Canada

    Antonio Lei

  3. Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146, Genova, Italy

    Stefano Vigni

Authors
  1. Jeffrey Hatley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antonio Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefano Vigni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeffrey Hatley.

Additional information

Publisher's Note

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

The second named author’s research is supported by the NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096. The third named author’s research is partially supported by PRIN 2017 “Geometric, algebraic and analytic methods in arithmetic”

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hatley, J., Lei, A. & Vigni, S. \(\Lambda \)-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves. manuscripta math. 167, 589–612 (2022). https://doi.org/10.1007/s00229-021-01283-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-021-01283-w

Mathematics Subject Classification

Search

Navigation