Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Mobile Device Pairing
 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

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


Author: Yoshihiro Ochi
Journal: Proc. Amer. Math. Soc. 134 (2006), 31-37
MSC (2000): Primary 11R23, 11G10
Posted: August 11, 2005
MathSciNet review: 2170540
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11R23, 11G10

Retrieve articles in all journals with MSC (2000): 11R23, 11G10

Additional Information

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

DOI: http://dx.doi.org/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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia