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Structure of fine Selmer groups in abelian p-adic Lie extensions

Published 21 Apr 2023 in math.NT | (2304.10938v2)

Abstract: This paper studies fine Selmer groups of elliptic curves in abelian $p$-adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic $\mathbb{Z}_p$-extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified.

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References (68)
  1. Aribam, C. S. On the μ𝜇\muitalic_μ-invariant of fine Selmer groups. J. Number Theory 135 (2014), 284–300.
  2. Bandini, A. Greenberg’s conjecture and capitulation in ℤpdsuperscriptsubscriptℤ𝑝𝑑\mathbb{Z}_{p}^{d}blackboard_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT-extensions. J. Number Theory 122, 1 (2007), 121–134.
  3. Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg’s p𝑝pitalic_p-rationality conjecture. J. Théor. Nombres Bordeaux 32, 1 (2020), 159–177.
  4. Bhave, A. Analogue of Kida’s formula for certain strongly admissible extensions. J. Number Theory 122, 1 (2007), 100–120.
  5. Higher Chern classes in Iwasawa theory. Amer. J. Math. 142, 2 (2020), 627–682.
  6. Brink, D. Prime decomposition in the anti-cyclotomic extension. Math. Comp. 76, 260 (2007), 2127–2138.
  7. Coates, J. Fragments of the GL2subscriptGL2{\rm GL}_{2}roman_GL start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Iwasawa theory of elliptic curves without complex multiplication. In Arithmetic theory of elliptic curves (Cetraro, 1997), vol. 1716 of Lecture Notes in Math. Springer, Berlin, 1999, pp. 1–50.
  8. Modules over Iwasawa algebras. J. Inst. Math. Jussieu 2, 1 (2003), 73–108.
  9. Galois cohomology of elliptic curves, vol. 88 of Tata Institute of Fundamental Research Lectures on Mathematics. Published by Narosa Publishing House, New Delhi; for the Tata Institute of Fundamental Research, Mumbai, 2000.
  10. Fine Selmer groups for elliptic curves with complex multiplication. In Algebra and number theory. Hindustan Book Agency, Delhi, 2005, pp. 327–337.
  11. Fine Selmer groups of elliptic curves over p𝑝pitalic_p-adic Lie extensions. Math. Ann. 331, 4 (2005), 809–839.
  12. On the Euler-Poincaré characteristics of finite dimensional p𝑝pitalic_p-adic Galois representations. Publ. Math. Inst. Hautes Études Sci., 93 (2001), 107–143.
  13. Analytic pro-p𝑝pitalic_p groups, second ed., vol. 61 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.
  14. The Iwasawa invariant μpsubscript𝜇𝑝\mu_{p}italic_μ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT vanishes for abelian number fields. Ann. of Math. (2) 109, 2 (1979), 377–395.
  15. Fujii, S. On a bound of λ𝜆\lambdaitalic_λ and the vanishing of μ𝜇\muitalic_μ of ℤpsubscriptℤ𝑝\mathbb{Z}_{p}blackboard_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-extensions of an imaginary quadratic field. J. Math. Soc. Japan 65, 1 (2013), 277–298.
  16. Fujii, S. On Greenberg’s generalized conjecture for CM-fields. J. Reine Angew. Math. 731 (2017), 259–278.
  17. Gras, G. Class field theory: From theory to practice. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. Translated from the French manuscript by Henri Cohen.
  18. Greenberg, R. Iwasawa theory for elliptic curves. In Arithmetic theory of elliptic curves (Cetraro, 1997), vol. 1716 of Lecture Notes in Math. Springer, Berlin, 1999, pp. 51–144.
  19. Greenberg, R. Introduction to Iwasawa theory for elliptic curves. Arithmetic algebraic geometry 9 (2001), 407–464.
  20. Greenberg, R. Iwasawa theory—past and present. In Class field theory—its centenary and prospect (Tokyo, 1998), vol. 30 of Adv. Stud. Pure Math. Math. Soc. Japan, Tokyo, 2001, pp. 335–385.
  21. Greenberg, R. Galois representations with open image. Ann. Math. Qué. 40, 1 (2016), 83–119.
  22. Howson, S. Iwasawa theory of Elliptic Curves for p𝑝pitalic_p-adic Lie extensions. PhD thesis, University of Cambridge, 1998.
  23. Howson, S. Euler characteristics as invariants of Iwasawa modules. Proc. London Math. Soc. (3) 85, 3 (2002), 634–658.
  24. Imai, H. A remark on the rational points of abelian varieties with values in cyclotomic Zpsubscript𝑍𝑝Z_{p}italic_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-extensions. Proc. Japan Acad. 51 (1975), 12–16.
  25. Iwasawa, K. On 𝐙lsubscript𝐙𝑙{\bf Z}_{l}bold_Z start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT-extensions of algebraic number fields. Ann. of Math. (2) 98 (1973), 246–326.
  26. Iwasawa, K. On the μ𝜇\muitalic_μ-invariants of Zℓsubscript𝑍ℓZ_{\ell}italic_Z start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT-extensions. In Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki. Kinokuniya, Tokyo, 1973, pp. 1–11.
  27. Jha, S. Fine Selmer group of Hida deformations over non-commutative p𝑝pitalic_p-adic Lie extensions. Asian J. Math. 16, 2 (2012), 353–365.
  28. On the Hida deformations of fine Selmer groups. J. Algebra 338 (2011), 180–196.
  29. Kato, K. Universal norms of p𝑝pitalic_p-units in some non-commutative Galois extensions. Doc. Math., Extra Vol. (2006), 551–565.
  30. Kim, B.-D. The plus/minus Selmer groups for supersingular primes. J. Aust. Math. Soc. 95, 2 (2013), 189–200.
  31. Kleine, S. Relative extensions of number fields and Greenberg’s generalised conjecture. Acta Arith. 174, 4 (2016), 367–392.
  32. Kobayashi, S. Iwasawa theory for elliptic curves at supersingular primes. Invent. Math. 152, 1 (2003), 1–36.
  33. Control theorems for fine Selmer groups. J. théor. Nombres Bordeaux (2022), accepted for publication.
  34. Lang, S. Elliptic functions, second ed., vol. 112 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1987. With an appendix by J. Tate.
  35. Conjectures de Greenberg et extensions pro-p𝑝pitalic_p-libres d’un corps de nombres. Manuscripta Math. 102, 2 (2000), 187–209.
  36. Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction. Forum Math. Sigma 7 (2019), Paper No. e25, 81.
  37. Mazur, B. Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18 (1972), 183–266.
  38. McCallum, W. G. Greenberg’s conjecture and units in multiple ℤpsubscriptℤ𝑝\mathbb{Z}_{p}blackboard_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-extensions. Amer. J. Math. 123, 5 (2001), 909–930.
  39. A cup product in the Galois cohomology of number fields. Duke Math. J. 120, 2 (2003), 269–310.
  40. Merel, L. Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124, 1-3 (1996), 437–449.
  41. Minardi, J. V. Iwasawa modules for ℤpdsubscriptsuperscriptℤ𝑑𝑝\mathbb{Z}^{d}_{p}blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-extensions of algebraic number fields. ProQuest LLC, Ann Arbor, MI, 1986. Thesis (Ph.D.)–University of Washington.
  42. Sur l’arithmétique des corps de nombres p𝑝pitalic_p-rationnels. In Séminaire de Théorie des Nombres, Paris 1987–88, vol. 81 of Progr. Math. Birkhäuser Boston, Boston, MA, 1990, pp. 155–200.
  43. Murty, V. K. Modular forms and the Chebotarev density theorem. II. In Analytic number theory (Kyoto, 1996), vol. 247 of London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 1997, pp. 287–308.
  44. Nekovář, J. Selmer complexes. Astérisque, 310 (2006), viii+559.
  45. Cohomology of number fields, second ed., vol. 323 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2008.
  46. Nguyen Quang Do, T. Analogues supérieurs du noyau sauvage. Sém. Théor. Nombres Bordeaux (2) 4, 2 (1992), 263–271.
  47. K2subscript𝐾2K_{2}italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT et conjecture de Greenberg dans les ℤpsubscriptℤ𝑝\mathbb{Z}_{p}blackboard_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-extensions multiples. J. Théor. Nombres Bordeaux 17, 2 (2005), 669–688.
  48. Residual supersingular Iwasawa theory and signed Iwasawa invariants. Rendiconti del Seminario Mathematico di Padova (2021), accepted for publication, DOI 10.4171/RSMUP/111.
  49. Ochi, Y. A remark on the pseudo-nullity conjecture for fine Selmer groups of elliptic curves. Comment. Math. Univ. St. Pauli 58, 1 (2009), 1–7.
  50. On the structure of Selmer groups over p𝑝pitalic_p-adic Lie extensions. J. Algebraic Geom. 11, 3 (2002), 547–580.
  51. Ozaki, M. Iwasawa invariants of ℤpsubscriptℤ𝑝\mathbb{Z}_{p}blackboard_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-extensions over an imaginary quadratic field. In Class field theory—its centenary and prospect (Tokyo, 1998), vol. 30 of Adv. Stud. Pure Math. Math. Soc. Japan, Tokyo, 2001, pp. 387–399.
  52. Perrin-Riou, B. Groupe de Selmer d’une courbe elliptique à multiplication complexe. Compositio Math. 43, 3 (1981), 387–417.
  53. The main conjecture for CM elliptic curves at supersingular primes. Ann. of Math. (2) 159, 1 (2004), 447–464.
  54. Ribet, K. Torsion points of abelian varieties in cyclotomic extensions. Enseign. Math 27 (1981), 315–319.
  55. Rubin, K. Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, 1999. AWS notes.
  56. Rubin, K. Euler systems, vol. 147 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2000. Hermann Weyl Lectures. The Institute for Advanced Study.
  57. Schneider, P. Über gewisse Galoiscohomologiegruppen. Math. Z. 168, 2 (1979), 181–205.
  58. Serre, J.-P. Abelian l𝑙litalic_l-adic representations and elliptic curves. W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute.
  59. Serre, J.-P. Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Études Sci. Publ. Math., 54 (1981), 323–401.
  60. Serre, J.-P. Galois cohomology. Springer-Verlag, Berlin, 1997. Translated from the French by Patrick Ion and revised by the author.
  61. Sharifi, R. T. On Galois groups of unramified pro-p𝑝pitalic_p extensions. Math. Ann. 342, 2 (2008), 297–308.
  62. Shekhar, S. Comparing the corank of fine Selmer group and Selmer group of elliptic curves. J. Ramanujan Math. Soc. 33, 2 (2018), 205–217.
  63. Takahashi, N. On Greenberg’s generalized conjecture for imaginary quartic fields. Int. J. Number Theory 17, 5 (2021), 1163–1173.
  64. Venjakob, O. On the structure theory of the Iwasawa algebra of a p𝑝pitalic_p-adic Lie group. J. Eur. Math. Soc. (JEMS) 4, 3 (2002), 271–311.
  65. Venjakob, O. A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory. J. Reine Angew. Math. 559 (2003), 153–191. With an appendix by Denis Vogel.
  66. Venjakob, O. On the Iwasawa theory of p𝑝pitalic_p-adic Lie extensions. Compositio Math. 138, 1 (2003), 1–54.
  67. Washington, L. C. Introduction to cyclotomic fields, 2 ed., vol. 83 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997.
  68. Wuthrich, C. Iwasawa theory of the fine Selmer group. J. Algebraic Geom. 16, 1 (2007), 83–108.
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