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A note on Selmer groups of abelian varieties over the trivializing extensions

Author(s): Yoshihiro Ochi
Journal: Proc. Amer. Math. Soc. 134 (2006), 31-37.
MSC (2000): Primary 11R23, 11G10
Posted: August 11, 2005
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Abstract: We prove that for any abelian variety $A$ defined over a number field $K$ that is not isogenous to a product of CM elliptic curves, the pontrjagin dual of the Selmer group of the abelian variety over the trivializing extension $K(A[p^\infty])$ has no nonzero pseudo-null submodules.

References:

[CG]
J. Coates and R. Greenberg, Kummer theory of abelian varieties over local fields, Invent. Math. 124 (1996), 129-174. MR 1369413 (97b:11079)

[Gr]
R. Greenberg, The structure of Selmer groups, Proceedings Natl. Acad. of Science 94 (1997), 11125-11128. MR 1491971 (98m:11123)

[Ma]
K. Matsuno, Finite $\Lambda$-submodules of Selmer groups of abelian varieties over cyclotomic $\mathbb{Z} _p$-extensions, Journal of Number Theory 99 (2003), 415-443. MR 1969183 (2004c:11098)

[Ja]
U. Jannsen, Iwasawa modules up to isomorphism, Advanced Studies in Pure Mathematics, vol. 17, Algebraic Number Theory, in honour of K. Iwasawa, pp. 171-207, 1989. MR 1097615 (93c:11095)

[NSW]
J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of Number Fields, Springer, 2000. MR 1737196 (2000j:11168)

[OV1]
Y. Ochi and O. Venjakob, On the structure of Selmer groups over $p$-adic Lie extensions, Journal of Algebraic Geometry 11 (2002), 547-580. MR 1894938 (2003m:11082)

[OV2]
Y. Ochi and O. Venjakov, On the ranks of Iwasawa modules over $p$-adic Lie extensions, Math. Proc. Camb. Phil. Soc. 135 (2003), 25-43. MR 1990830 (2004d:11107)
[Pe]
B. Perrin-Riou, Groupe de Selmer d'une courbe elliptique a multiplication complexe, Compo. Math., 43 (1981), 387-417. MR 0632436 (83i:14031)

[Ri]
K. Ribet, Division fields of abelian varieties with complex multiplication, Memoires de la SMF 1980, pp. 75-94. MR 0608640 (83e:14029a)

[Se]
J-P. Serre, Collected Papers IV, Springer. MR 1730973 (2001e:01037)

[ST]
J-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492-517. MR 0236190 (38:4488)

[Ve]
O. Venjakob, On the structure theory of the Iwasawa algebra of a $p$-adic Lie group, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 3, 271-311. MR 1924402 (2004h:16029)

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Additional Information:

Yoshihiro Ochi
Affiliation: School of Science and Engineering, Tokyo Denki University, Tokyo, 101--8457, Japan
Email: ochi@u.dendai.ac.jp

DOI: 10.1090/S0002-9939-05-08292-4
PII: S 0002-9939(05)08292-4
Keywords: Selmer groups, abelian varieties, Iwasawa theory.
Received by editor(s): August 11, 2004
Posted: August 11, 2005
Communicated by: David E. Rohrlich
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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