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Akashi series of Selmer groups

Published online by Cambridge University Press:  04 May 2011

SARAH LIVIA ZERBES
SARAH LIVIA ZERBES*
Affiliation:
Department of Mathematics, Harrison Building, University of Exeter, Exeter EX4 4QF. e-mail: s.zerbes@exeter.ac.uk
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Abstract

We study the Selmer group of an elliptic curve over an admissible p-adic Lie extension of a number field F. We give a formula for the Akashi series attached to this module, in terms of the corresponding objects for the cyclotomic ℤp-extension and certain correction terms. This extends our earlier work [16], in particular since it applies to elliptic curves having split multiplicative reduction at some primes above p, in which case the Akashi series can have additional zeros.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 2 , September 2011 , pp. 229 - 243
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Balister, P. and Howson, S.Note on Nakayama's lemma for compact Λ-modules. Asian J. Math. 1, no. 2 (1997), 224–229.CrossRefGoogle Scholar
[2]Burns, D. and Venjakob, O. On the leading terms of zeta isomorphisms and p-adic L-functions in non-commutative Iwasawa theory. In John H. Coates' Sixtieth Birthday, Doc. Math. Extra Vol. (2006), pp. 165–209 (electronic).CrossRefGoogle Scholar
[3]Coates, J. Fragments of the GL2 Iwasawa theory of elliptic curves without complex multiplication. In Arithmetic theory of elliptic curves (Cetraro, 1997), Lecture Notes in Math. vol. 1716 (Springer, 1999), pp. 1–50.CrossRefGoogle Scholar
[4]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. 101 (2005), 163–208.CrossRefGoogle Scholar
[5]Coates, J. and Greenberg, R.Kummer theory for abelian varieties over local fields. Invent. Math. 124 (1996), 129–174.CrossRefGoogle Scholar
[6]Coates, J. and Howson, S.Euler characteristics and elliptic curves. II. J. Math. Soc. Japan 53 (2001), 175–235.CrossRefGoogle Scholar
[7]Coates, J., Schneider, P. and Sujatha, R. Links between cyclotomic and GL2 Iwasawa theory. In Kazuya Kato's fiftieth birthday, Doc. Math. Extra Vol. (2003), pp. 187–215 (electronic).CrossRefGoogle Scholar
[8]Cremona, J. E.Algorithms for Modular Elliptic Curves (second edition) (Cambridge University Press, 1997).Google Scholar
[9]Hachimori, Y. and Venjakob, O. Completely faithful Selmer groups over Kummer extensions. In Kazuya Kato's fiftieth birthday, Doc. Math. Extra Vol. (2003), pp. 443–478 (electronic).CrossRefGoogle Scholar
[10]Imai, H.A remark on the rational points of abelian varieties with values in cyclotomic Zp-extensions. Proc. Japan Acad. 51 (1975), 12–16.Google Scholar
[11]Kato, K.P-adic Hodge theory and values of zeta functions of modular forms. In Cohomologies p-adiques et applications arithmetiques (III), Astérisque 295 (2004), pp. 117–290.Google Scholar
[12]Serre, J.-P.Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Invent. Math. 15 (1972), 259–331.CrossRefGoogle Scholar
[13]Zábrádi, G.Characteristic elements, pairings and functional equations over the false Tate curve extension. Math. Proc. Camb. Phil. Soc. 144 (2008), 535–574.CrossRefGoogle Scholar
[14]Zábrádi, G.Pairings and functional equations over the GL2-extension. Proc. London Math. Soc. 101 (2010), 893–930.CrossRefGoogle Scholar
[15]Zerbes, S. L.Selmer groups over p-adic Lie extensions I. J. London Math. Soc. 70 (2004), 586–608.CrossRefGoogle Scholar
[16]Zerbes, S. L.Generalized Euler characteristics of Selmer groups. Proc. London Math. Soc. 98 (2009), 775–796.CrossRefGoogle Scholar