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THE PLUS/MINUS SELMER GROUPS FOR SUPERSINGULAR PRIMES

Published online by Cambridge University Press:  07 June 2013

BYOUNG DU KIM
BYOUNG DU KIM*
Affiliation:
Victoria University of Wellington, Wellington 6140, New Zealand email [email protected]
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Abstract

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Suppose that an elliptic curve $E$ over $ \mathbb{Q} $ has good supersingular reduction at $p$. We prove that Kobayashi’s plus/minus Selmer group of $E$ over a ${ \mathbb{Z} }_{p} $-extension has no proper $\Lambda $-submodule of finite index under some suitable conditions, where $\Lambda $ is the Iwasawa algebra of the Galois group of the ${ \mathbb{Z} }_{p} $-extension. This work is analogous to Greenberg’s result in the ordinary reduction case.

Keywords

elliptic curvesIwasawa theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 95 , Issue 2 , October 2013 , pp. 189 - 200
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Coates, J. and Greenberg, R., ‘Kummer theory for abelian varieties over local fields’, Invent. Math. 124 (1996), 129–174.CrossRefGoogle Scholar
Greenberg, R., ‘Iwasawa theory for elliptic curves’, in: Arithmetic Theory of Elliptic Curves (Cetraro, 1997), Lecture Notes in Mathematics, 1716 (Springer, Berlin, 1999), 51–144.CrossRefGoogle Scholar
Hazewinkel, M., ‘On norm maps for one dimensional formal groups. I. The cyclotomic$\Gamma $-extension’, J. Algebra 32 (1974), 89–108.CrossRefGoogle Scholar
Hazewinkel, M., ‘On norm maps for one dimensional formal groups III’, Duke Math. J. 44 (2) (1977), 305–314.CrossRefGoogle Scholar
Iovita, A. and Pollack, R., ‘Iwasawa theory of elliptic curves at supersingular primes over towers of extensions of number fields’, J. reine angew. Math. 598 (2006), 71–103.Google Scholar
Kim, B. D., ‘The parity conjecture for elliptic curves at supersingular reduction primes’, Compositio Math. 143 (2007), 47–72.CrossRefGoogle Scholar
Kim, B. D., ‘The Iwasawa invariants of the plus/minus Selmer groups’, Asian J. Math. 13 (2) (2009), 181–190.CrossRefGoogle Scholar
Kobayashi, S., ‘Iwasawa theory for elliptic curves at supersingular primes’, Invent. Math. 152 (1) (2003), 1–36.Google Scholar
Lang, S., Algebra, 3rd edn (Springer, New York, 2002).CrossRefGoogle Scholar
Mazur, B., ‘Rational points of abelian varieties with values in towers of number fields’, Invent. Math. 18 (1972), 183–266.CrossRefGoogle Scholar
Pollack, R., ‘On the p-adic L-function of a modular form at a supersingular prime’, Duke Math. J. 118 (3) (2003), 523–558.CrossRefGoogle Scholar
Silverman, J. H., The Arithmetic of Elliptic Curves, 2nd edn. Graduate Texts in Mathematics, 106 (Springer, New York, 2009).CrossRefGoogle Scholar