A note on Selmer groups of abelian varieties over the trivializing extensions
HTML articles powered by AMS MathViewer
- by Yoshihiro Ochi PDF
- Proc. Amer. Math. Soc. 134 (2006), 31-37 Request permission
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
- J. Coates and R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), no. 1-3, 129–174. MR 1369413, DOI 10.1007/s002220050048
- Ralph Greenberg, The structure of Selmer groups, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 21, 11125–11128. Elliptic curves and modular forms (Washington, DC, 1996). MR 1491971, DOI 10.1073/pnas.94.21.11125
- Kazuo Matsuno, Finite $\Lambda$-submodules of Selmer groups of abelian varieties over cyclotomic $\Bbb Z_p$-extensions, J. Number Theory 99 (2003), no. 2, 415–443. MR 1969183, DOI 10.1016/S0022-314X(02)00078-1
- Uwe Jannsen, Iwasawa modules up to isomorphism, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 171–207. MR 1097615, DOI 10.2969/aspm/01710171
- Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR 1737196
- Yoshihiro Ochi and Otmar Venjakob, On the structure of Selmer groups over $p$-adic Lie extensions, J. Algebraic Geom. 11 (2002), no. 3, 547–580. MR 1894938, DOI 10.1090/S1056-3911-02-00297-7
- Yoshihiro Ochi and Otmar Venjakob, On the ranks of Iwasawa modules over $p$-adic Lie extensions, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 1, 25–43. MR 1990830, DOI 10.1017/S0305004102006564
- Bernadette Perrin-Riou, Groupe de Selmer d’une courbe elliptique à multiplication complexe, Compositio Math. 43 (1981), no. 3, 387–417 (French). MR 632436
- K. A. Ribet, Division fields of abelian varieties with complex multiplication, Mém. Soc. Math. France (N.S.) 2 (1980/81), 75–94. MR 608640
- Jean-Pierre Serre, Œuvres. Collected papers. IV, Springer-Verlag, Berlin, 2000 (French). 1985–1998. MR 1730973, DOI 10.1007/978-3-642-41978-2
- Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
- Otmar 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, DOI 10.1007/s100970100038
Additional Information
- Yoshihiro Ochi
- Affiliation: School of Science and Engineering, Tokyo Denki University, Tokyo, 101–8457, Japan
- Email: ochi@u.dendai.ac.jp
- Received by editor(s): August 11, 2004
- Published electronically: August 11, 2005
- Communicated by: David E. Rohrlich
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 31-37
- MSC (2000): Primary 11R23, 11G10
- DOI: https://doi.org/10.1090/S0002-9939-05-08292-4
- MathSciNet review: 2170540