Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-16T22:35:44.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative reduction

Published online by Cambridge University Press:  29 October 2012

CHERN–YANG LEE
CHERN–YANG LEE*
Affiliation:
The School of Mathematical Sciences, The University of Nottingham, Nottingham NG7 2RD. e-mail: chern-yang.lee@nottingham.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the compact p∞-Selmer Iwasawa module X(E/F∞) of an elliptic curve E over a False Tate curve extension F∞, where E is defined over ℚ, having multiplicative reduction at the odd prime p. We investigate the p∞-Selmer rank of E over intermediate fields and give the best lower bound of its growth under certain parity assumption on X(E/F∞), assuming this Iwasawa module satisfies the H(G)-Conjecture proposed by Coates–Fukaya–Kato–Sujatha–Venjakob.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 2 , March 2013 , pp. 303 - 324
Copyright
Copyright © Cambridge Philosophical Society 2012

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

REFERENCES

[1]Coates, J., Fukaya, T., Kato, K. and Sujathan, R.Root numbers, Selmer groups and non-commutative Iwasawa theory. J. Algebraic Geom. 19 (2010), no. 1, 19–97.CrossRefGoogle Scholar
[2]Coates, J., Fukaya, T., Kato, K., Sujatha, R. and Venjakob, O. The GL2 main conjecture for elliptic curves without complex multiplication. Publ. Math. Inst. Hautes Études Sci. (2005), no. 101, 163–208.Google Scholar
[3]Coates, J. and Greenberg, R.Kummer theory for abelian varieties over local fields. Invent. Math. 124 (1996), no. 1-3, 129–174.CrossRefGoogle Scholar
[4]Dokchitser, T. and Dokchitser, V.Self-duality of Selmer groups. Math. Proc. Camb. Phils. Soc. 146 (2009), no. 2, 257–267.CrossRefGoogle Scholar
[5]Dokchitser, V.Root numbers of non-abelian twists of elliptic curves. Proc. London Math. Soc. 3 (2005), no. 91, 300–324.CrossRefGoogle Scholar
[6]Greenberg, R.Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, 1997). Lecture Notes in Math. vol. 1716. (Springer, Berlin, 1999), pp. 51–144.Google Scholar
[7]Guo, L.On a generalization of Tate dualities with application to Iwasawa theory. Compositio Math. 85 (1993), no. 2, 125–161.Google Scholar
[8]Hachimori, Y. and Venjakob, O. Completely faithful selmer groups over kummer extensions. Doc. Math. Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 443–478.CrossRefGoogle Scholar
[9]Howson, S.Euler characteristics as invariants of Iwasawa modules. Proc. London Math. Soc. (3) 85 (2002), no. 3, 634–658.CrossRefGoogle Scholar
[10]Jones, J. Iwasawa L-functions and the mysterious -invariant, p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991). Contemp. Math. vol. 165, (Amer. Math. Soc., Providence, RI, 1994), pp. 63–70.Google Scholar
[11]Lang, S.Algebra, third ed., Graduate Texts in Mathematics, vol. 211 (Springer-Verlag, New York, 2002).CrossRefGoogle Scholar
[12]Ribet, K.Torsion points of abelian varieties in cyclotomic extensions. Lenseignement Mathematiqu. 27 (1981), 315–319.Google Scholar
[13]Schinzel, A.Abelian binomials, power residues and exponential congruences. Acta Arith. 32 (1977), no. 3, 245–274.CrossRefGoogle Scholar
[14]Vélez, W. Y.The factorization of p in Q(a1/pk) and the genus field of Q(a1/n). Tokyo J. Math. 11 (1988), no. 1, 1–19.Google Scholar
[15]Viviani, F.Ramification groups and Artin conductors of radical extensions of ℚ. J. Théor. Nombres Bordeau. 16 (2004), no. 3, 779–816.CrossRefGoogle Scholar