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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: [email protected]
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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

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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