The torsion of 𝑝-ramified Iwasawa modules II

Fujii, Satoshi

doi:10.1090/S0002-9939-07-09029-6

The torsion of $p$-ramified Iwasawa modules II
HTML articles powered by AMS MathViewer

by Satoshi Fujii PDF

Proc. Amer. Math. Soc. 136 (2008), 461-466 Request permission

Abstract:

In this article we prove the existence of a non-trivial torsion of the $3$-ramified Iwasawa mocule over the $\mathbb {Z}_3^2$-extension of an imaginary quadratic field.

References

Andrea Bandini, Greenberg’s conjecture for ${\Bbb Z}^d_p$-extensions, Acta Arith. 108 (2003), no. 4, 357–368. MR 1979904, DOI 10.4064/aa108-4-5
H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields, Bull. London Math. Soc. 1 (1969), 345–348. MR 254010, DOI 10.1112/blms/1.3.345
H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405–420. MR 491593, DOI 10.1098/rspa.1971.0075
S. Fujii, The torsion of $p$-ramified Iwasawa modules, submitted.
Ralph Greenberg, On the structure of certain Galois groups, Invent. Math. 47 (1978), no. 1, 85–99. MR 504453, DOI 10.1007/BF01609481
Ralph Greenberg, Iwasawa theory—past and present, Class field theory—its centenary and prospect (Tokyo, 1998) Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001, pp. 335–385. MR 1846466, DOI 10.2969/aspm/03010335
Jin Nakagawa and Kuniaki Horie, Elliptic curves with no rational points, Proc. Amer. Math. Soc. 104 (1988), no. 1, 20–24. MR 958035, DOI 10.1090/S0002-9939-1988-0958035-0
Yasutaka Ihara, Profinite braid groups, Galois representations and complex multiplications, Ann. of Math. (2) 123 (1986), no. 1, 43–106. MR 825839, DOI 10.2307/1971352
T. Itoh, Pseudo-null Iwasawa modules for $\mathbb {Z}_2^2$-extensions, submitted for publication.
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
Arthur Lannuzel and Thong Nguyen Quang Do, Conjectures de Greenberg et extensions pro-$p$-libres d’un corps de nombres, Manuscripta Math. 102 (2000), no. 2, 187–209 (French, with English summary). MR 1771439, DOI 10.1007/s002291020187
J. Minardi, Iwasawa modules for $\mathbb {Z}_p^d$-extensions of algebraic number fields, Thesis (1986), Washington University.
A. Movahhedi, Sur les $p$-extensions des corps $p$-rationnels, Math. Nachr. 149 (1990), 163–176 (French). MR 1124802, DOI 10.1002/mana.19901490113
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
T. Nguyễn-Quang-Đỗ, Formations de classes et modules d’Iwasawa, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 167–185 (French). MR 756093, DOI 10.1007/BFb0099451
Hisao Taya, Iwasawa invariants and class numbers of quadratic fields for the prime $3$, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1285–1292. MR 1641133, DOI 10.1090/S0002-9939-99-05177-1
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7