Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T19:33:57.070Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounding the Iwasawa invariants of Selmer groups

Part of: Theory of modules and ideals Algebraic number theory: global fields Arithmetic algebraic geometry

Published online by Cambridge University Press:  29 June 2020

Sören Kleine
Sören Kleine*
Affiliation:
Institut für Theoretische Informatik, Mathematik und Operations Research, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, D-85577 Neubiberg, Germany
*
e-mail: soeren.kleine@unibw.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

Keywords

Fukuda moduleSelmer groupMordell–Weil rankfine Selmer group

MSC classification

Primary: 11R23: Iwasawa theory
Secondary: 11G05: Elliptic curves over global fields 11G10: Abelian varieties of dimension $> 1$ 13C12: Torsion modules and ideals
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 5 , October 2021 , pp. 1390 - 1422
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bertolini, M. and Darmon, H., Iwasawa’s main conjecture for elliptic curves over anticyclotomic ${\mathbb{Z}}_p$ -extensions. Ann. of Math. (2) 162(2005), 1–64. http://dx.doi.org/10.4007/annals.2005.162.1 CrossRefGoogle Scholar
Coates, J. and Sujatha, R., Fine Selmer groups of elliptic curves over $p$ -adic Lie extensions. Math. Ann. 331(2005), 809–839. http://dx.doi.org/10.1007/s00208-004-0609-z CrossRefGoogle Scholar
Coates, J. and Sujatha, R., Galois cohomology of elliptic curves . 2nd ed., Narosa Publishing House/Published for the Tata Institute of Fundamental Research, Mumbai, 2010.Google Scholar
Darmon, H. and Iovita, A., The anticyclotomic main conjecture for elliptic curves at supersingular primes . J. Inst. Math. Jussieu 7(2008), 291–325. http://dx.doi.org/10.1017/S1474748008000042 CrossRefGoogle Scholar
Fukuda, T., Remarks on ${\mathbb{Z}}_p$ -extensions of number fields. Proc. Jpn. Acad. Ser. A Math. Sci. 70(1994), 264–266.CrossRefGoogle Scholar
Greenberg, R., The Iwasawa invariants of $\varGamma$ -extensions of a fixed number field. Amer. J. Math. 95(1973), 204–214. http://dx.doi.org/10.2307/2373652 CrossRefGoogle Scholar
Greenberg, R., Introduction to Iwasawa theory for elliptic curves . In: Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., 9, Amer. Math. Soc., Providence, RI, 2001. http://dx.doi.org/10.1090/pcms/009/06 Google Scholar
Greenberg, R., Iwasawa theory—past and present . In: Class field theory—its centenary and prospect (Tokyo, 1998), Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo, 2001. http://dx.doi.org/10.2969/aspm/03010335 Google Scholar
Greenberg, R., Galois theory for the Selmer group of an abelian variety . Compos. Math. 136(2003), 255–297. https://doi.org/10.1023/A:1023251032273 CrossRefGoogle Scholar
Imai, H., A remark on the rational points of Abelian varieties with values in cyclotomic ${\mathbb{Z}}_p$ -extensions. Proc. Jpn. Acad. 51(1975), 12–16.Google Scholar
Iovita, A. and Pollack, R., Iwasawa theory of elliptic curves at supersingular primes over ${\mathbb{Z}}_p$ -extensions of number fields. J. Reine Angew. Math. 598(2006), 71–103. http://dx.doi.org/10.1515/CRELLE.2006.069 Google Scholar
Iwasawa, K., On ${\mathbb{Z}}_{\ell}$ -extensions of algebraic number fields. Ann. Math. (2) 98(1973), 246–326.CrossRefGoogle Scholar
Iwasawa, K., On the $\mu$ -invariants of ${\mathbb{Z}}_{\ell }$ -extensions. In: Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 1–11.Google Scholar
Kato, K., $p$ -adic Hodge theory and values of zeta functions of modular forms. Cohomologies $p$ -adiques et applications arithmétiques (III). Astérisque 295(2004), 117–290.Google Scholar
Katz, N. M. and Lang, S., Finiteness theorems in geometric classfield theory (with an appendix by K. A. Ribet). Enseign. Math. (2) 27(1981), 285–319.Google Scholar
Kleine, S., Local behavior of Iwasawa’s invariants . Int. J. Number Theory 13(2017), 1013–1036. https://doi.org/10.1142/S1793042117500543 CrossRefGoogle Scholar
Kleine, S., Local behaviour of generalised Iwasawa invariants . Ann. Math. Qué. 43(2019), 305–339. http://dx.doi.org/10.1007/s40316-018-0106-5 CrossRefGoogle Scholar
Kleine, S., Generalised Iwasawa invariants and the growth of class numbers. Forum Math., 2020. https://doi.org/10.1515/forum-2019-0119 CrossRefGoogle Scholar
Kurihara, M., On the Tate-Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction . I. Invent. Math. 149(2002), 195–224. http://dx.doi.org/10.1007/s002220100206 CrossRefGoogle Scholar
Lei, A. and Ponsinet, G., On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions . Proc. Am. Math. Soc. Ser. B 7(2020), 1–16. http://dx.doi.org/10.1090/bproc/43 CrossRefGoogle Scholar
Manin, Y. I., Cyclotomic fields and modular curves . Russ. Math. Surv. 26(1971), 7–71.CrossRefGoogle Scholar
Matsuno, K., Mordell-Weil ranks of elliptic curves in the cyclotomic ${\mathbb{Z}}_2$ -extension of the rationals. Int. J. Number Theory 13(2017), 429–438. http://dx.doi.org/10.1142/S1793042117500257 CrossRefGoogle Scholar
Mazur, B., Rational points of Abelian varieties with values in towers of number fields . Invent. Math. 18(1972), 183–266. http://dx.doi.org/10.1007/BF01389815 CrossRefGoogle Scholar
Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of number fields . 2nd ed., Grundlehren der Mathematischen Wissenschaften, 323, Springer-Verlag, Berlin, 2008. http://dx.doi.org/10.1007/978-3-540-37889-1 Google Scholar
Pollack, R., An algebraic version of a theorem of Kurihara . J. Number Theory 110(2005), 164–177. http://dx.doi.org/10.1016/j.jnt.2003.10.008 CrossRefGoogle Scholar
Rohrlich, D. E., On $L$ -functions of elliptic curves and cyclotomic towers. Invent. Math. 75(1984), 409–423. http://dx.doi.org/10.1007/BF01388636 CrossRefGoogle Scholar
Rubin, K., The one-variable main conjecture for elliptic curves with complex multiplication . In: $L$ -functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., 153, Cambridge Univ. Press, Cambridge, UK, 1991, pp. 353–371. http://dx.doi.org/10.1017/CBO9780511526053.015 Google Scholar
Skinner, C. and Urban, E., The Iwasawa main conjectures for ${\text{GL}}_2$ . Invent. Math. 195(2014), 1–277. http://dx.doi.org/10.1007/s00222-013-0448-1 CrossRefGoogle Scholar
Washington, L. C., Introduction to cyclotomic fields . 2nd ed., Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997. http://dx.doi.org/10.1007/978-1-4612-1934-7 Google Scholar
Wingberg, K., On the rational points of abelian varieties over ${\mathbb{Z}}_p$ -extensions of number fields. Math. Ann. 279(1987), 9–24. http://dx.doi.org/10.1007/BF01456190 CrossRefGoogle Scholar
Wuthrich, C., Iwasawa theory of the fine Selmer group . J. Algebr. Geom. 16(2007), 83–108. http://dx.doi.org/10.1090/S1056-3911-06-00436-X CrossRefGoogle Scholar
Zarhin, Y. G., Torsion of abelian varieties, Weil classes and cyclotomic extensions . Math. Proc. Cambridge Philos. Soc. 126(1999), 1–15. http://dx.doi.org/10.1017/S0305004198003235 CrossRefGoogle Scholar