Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-19T19:37:52.926Z Has data issue: false hasContentIssue false
  • English
  • Français

On the cohomology of Kobayashi’s plus/minus norm groups and applications

Part of: Algebraic number theory: local and $p$-adic fields Algebraic number theory: global fields Arithmetic algebraic geometry

Published online by Cambridge University Press:  11 May 2021

MENG FAI LIM
MENG FAI LIM*
Affiliation:
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Hubei, Wuhan No. 152 Luoyu Road, 430079, P.R.China. e-mail: limmf@mail.ccnu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

The plus and minus norm groups are constructed by Kobayashi as subgroups of the formal group of an elliptic curve with supersingular reduction, and they play an important role in Kobayashi’s definition of the signed Selmer groups. In this paper, we study the cohomology of these plus and minus norm groups. In particular, we show that these plus and minus norm groups are cohomologically trivial. As an application of our analysis, we establish certain (quasi-)projectivity properties of the non-primitive mixed signed Selmer groups of an elliptic curve with good reduction at all primes above p. We then build on these projectivity results to derive a Kida formula for the signed Selmer groups under a slight weakening of the usual µ = 0 assumption, and study the integrality property of the characteristic element attached to the signed Selmer groups.

MSC classification

Primary: 11R23: Iwasawa theory
Secondary: 11G05: Elliptic curves over global fields 11S25: Galois cohomology
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 1 , July 2022 , pp. 1 - 24
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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

Ahmed, S. and Lim, M. F.. On the Euler characteristics of signed Selmer groups. Bull. Aust. Math. Soc. 101 (2020), no. 2, 238–246.CrossRefGoogle Scholar
Ahmed, S. and Lim, M. F.. On the signed Selmer groups of congruent elliptic curves with semistable reduction at all primes above p. Acta Arithmetica 197 (2021), no. 4, 353–377.CrossRefGoogle Scholar
Büyükboduk, K. and Lei, A.. Integral Iwasawa theory of Galois representations for non-ordinary primes. Math. Z. 286 (2017), no. 1–2, 361–398.CrossRefGoogle Scholar
Coates, J., Fukaya, T., Kato, K., Sujatha, R. and Venjakob, O.. The GL 2 main conjecture for elliptic curves without complex multiplication. Publ. Math. Inst. Hautes Études Sci. No. 101, (2005), 163–208.CrossRefGoogle Scholar
Coates, J. and Greenberg, R.. Kummer theory for abelian varieties over local fields. Invent. Math. 124 (1996), no. 1–3, 129–174.CrossRefGoogle Scholar
Ellerbrock, N. and Nickel, A.. On formal groups and Tate cohomology in local fields. Acta Arith. 182 (2018), no. 3, 285–299.CrossRefGoogle Scholar
Greenberg, R.. Iwasawa theory for p-adic representations, in: Algebraic number theory, 97–137. Adv. Stud. Pure Math., 17 (Academic Press, Boston, MA, 1989).Google Scholar
Greenberg, R.. Iwasawa theory for elliptic curves, in: Arithmetic theory of elliptic curves (Cetraro, 1997), 51–144, Lecture Notes in Math., 1716 (Springer, Berlin, 1999).Google Scholar
Greenberg, R.. Iwasawa theory, projective modules and modular representations. Mem. Amer. Math. Soc. 211 (2011), no. 992, vi+185 pp.Google Scholar
Hachimori, Y. and Matsuno, K.. An analogue of Kida’s formula for the Selmer groups of elliptic curves. J. Algebraic Geom. 8 (1999), no. 3, 581–601.Google Scholar
Hachimori, Y. and Sharifi, R.. On the failure of pseudo-nullity of Iwasawa modules. J. Algebraic Geom. 14 (2005), no. 3, 567–591.CrossRefGoogle Scholar
Hatley, J. and Lei, A.. Arithmetic properties of signed Selmer groups at non-ordinary primes. Ann. Inst. Fourier (Grenoble) 69 (2019), no. 3, 1259–1294.CrossRefGoogle Scholar
Iwasawa, K.. Riemann–Hurwitz formula and p-adic Galois representations for number fields. Tohoku Math. J. (2) 33 (1981), no. 2, 263–288.CrossRefGoogle Scholar
Kakde, M.. Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases. J. Algebraic Geom. 20 (2011), no. 4, 631–683.CrossRefGoogle Scholar
Kakde, M.. The main conjecture of Iwasawa theory for totally real fields. Invent. Math. 193 (2013), no. 3, 539–626.CrossRefGoogle Scholar
Kato, K.. p-adic Hodge theory and values of zeta functions of modular forms, in: Cohomologies p-adiques et applications arithmétiques. III. Astérisque 295 (2004), ix, pp. 117–290.Google Scholar
Kim, B. D.. The parity conjecture for elliptic curves at supersingular reduction primes. Compos. Math. 143 (2007), no. 1, 47–72.CrossRefGoogle Scholar
Kim, B. D.. The plus/minus Selmer groups for supersingular primes. J. Aust. Math. Soc. 95 (2) (2013) 189–200.CrossRefGoogle Scholar
Kim, B. D.. Signed-Selmer groups over the $${\mathbb Z}_p^2$$-extension of an imaginary quadratic field. Canad. J. Math. 66 (2014), no. 4, 826–843.CrossRefGoogle Scholar
Kitajima, T. and Otsuki, R.. On the plus and the minus Selmer groups for elliptic curves at supersingular primes. Tokyo J. Math. 41 (2018), no. 1, 273–303.CrossRefGoogle Scholar
Kobayashi, S.. Iwasawa theory for elliptic curves at supersingular primes. Invent. Math. 152 (2003), no. 1, 1–36.CrossRefGoogle Scholar
Lei, A.. Non-commutative p-adic L-functions for supersingular primes. Int. J. Number Theory 8 (2012), no. 8, 1813–1830 CrossRefGoogle Scholar
Lei, A. and Lim, M. F.. Mordell–Weil ranks and Tate–Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions, arXiv:1911.10643 [math.NT].Google Scholar
Lei, A. and Lim, M. F.. Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p, accepted for publication in J. Theor. Nombres Bordeaux.Google Scholar
Lei, A. and Zerbes, S.. Signed Selmer groups over p-adic Lie extensions. J. Théor. Nombres Bordeaux 24 (2012), no. 2, 377–403.CrossRefGoogle Scholar
Mazur, B.. Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18 (1972), 183–266.CrossRefGoogle Scholar
J. Neukirch A. Schmidt and K. Wingberg. Cohomology of Number Fields, 2nd edn., Grundlehren Math. Wiss. 323 (Springer-Verlag, Berlin, 2008).CrossRefGoogle Scholar
Nichifor, A. and Palvannan, B.. On free resolutions of Iwasawa modules. Doc. Math. 24 (2019), 609–662.Google Scholar
Noether, E.. Normalbasis bei Körpern ohne höhere Verzweigung. J. Reine Angew. Math. 167 (1932), 147–152.CrossRefGoogle Scholar
Pollack, R. and Weston, T.. Kida’s formula and congruences. Doc. Math. (2006), Extra Vol., 615–630.Google Scholar
Reiner, I.. Maximal orders. Corrected reprint of the 1975 original. With a foreword by M. J. Taylor. London Math. Soc. Monog. New Series, 28 (The Clarendon Press, Oxford University Press, Oxford, 2003), xiv+395 pp.Google Scholar
Witte, M.. On a localisation sequence for the K-theory of skew power series rings. J. K-Theory 11 (2013), no. 1, 125–154.CrossRefGoogle Scholar