Proceedings of the American Mathematical Society

Remark on elliptic units in a $\mathbb{Z}_p$ -extension of an imaginary quadratic field

Author(s): Tsuyoshi Itoh
Journal: Proc. Amer. Math. Soc. 137 (2009), 473-478.
MSC (2000): Primary 11R23; Secondary 11G16
Posted: August 20, 2008
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Abstract: We shall study the group of units modulo the group of elliptic units in a $\mathbb{Z}_p$ -extension of an imaginary quadratic field.

1.: J. Coates and S. Lichtenbaum : On -adic zeta functions, Ann. of Math. (2) 98 (1973), 498-550. MR 0330107 (48:8445)
2.: J. Coates and W. Sinnott : Integrality properties of the values of partial zeta functions, Proc. London Math. Soc. (3) 34 (1977), 365-384. MR 0439815 (55:12697)
3.: T. Fukuda and K. Komatsu : Noncyclotomic $\mathbb{Z}_p$ -extensions of imaginary quadratic fields, Experiment. Math. 11 (2002), 469-475. MR 1969639 (2004g:11097)
4.: R. Gillard : Unités elliptiques et unités cyclotomiques, Math. Ann. 243 (1979), 181-189. MR 543728 (81k:12007)
5.: R. Greenberg : On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263-284. MR 0401702 (53:5529)
6.: R. Greenberg : Iwasawa theory--past and present, Class field theory--its centenary and prospect, Adv. Stud. Pure Math., 30, 335-385, Math. Soc. Japan, Tokyo, 2001. MR 1846466 (2002f:11152)
7.: T. Itoh and K. Komatsu : On the group of modular units and the ideal class group, J. Number Theory 123 (2007), 193-203. MR 2295439
8.: K. Iwasawa : On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan 16 (1964), 42-82. MR 0215811 (35:6646)
9.: K. Iwasawa : On $\mathbb{Z}_l$ -extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246-326. MR 0349627 (50:2120)
10.: D. Kersey : Modular units inside cyclotomic units, Ann. of Math. (2) 112 (1980), 361-380. MR 592295 (82h:12006)
11.: J-M. Kim, S. Bae, and I-S. Lee : Cyclotomic units in $\mathbb{Z}_p$ -extensions, Israel J. Math. 75 (1991), 161-165. MR 1164588 (93i:11120)
12.: D. S. Kubert and S. Lang : Modular units inside cyclotomic units, Bull. Soc. Math. France 107 (1979), 161-178. MR 545170 (81k:12006)
13.: D. S. Kubert and S. Lang : Modular units, Springer-Verlag, New York, Heidelberg, Berlin, 1981. MR 648603 (84h:12009)
14.: H. Oukhaba : Index formulas for ramified elliptic units, Compositio Math. 137 (2003), 1-22. MR 1981934 (2004b:11085)
15.: M. Ozaki : On the cyclotomic unit group and the ideal class group of a real abelian number field, J. Number Theory 64 (1997), 211-222. MR 1453211 (98c:11121)
16.: M. Ozaki : On the cyclotomic unit group and the ideal class group of a real abelian number field. II, J. Number Theory 64 (1997), 223-232. MR 1453211 (98c:11121)
17.: M. Ozaki : Iwasawa invariants of $\mathbb{Z}_p$ -extensions over an imaginary quadratic field, Class field theory--its centenary and prospect, Adv. Stud. Pure Math., 30, 387-399, Math. Soc. Japan, Tokyo, 2001. MR 1846467 (2002e:11147)
18.: V. P. Snaith : Algebraic -groups as Galois modules, Birkhäuser Verlag, Basel, Boston, Berlin, 2002. MR 1897817 (2003c:11149)
19.: H. M. Stark : -functions at . IV, Adv. in Math. 35 (1980), 197-235. MR 563924 (81f:10054)
20.: L. C. Washington : Introduction to cyclotomic fields, 2nd edition, Springer-Verlag, New York, Berlin, Heidelberg, 1997. MR 1421575 (97h:11130)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11R23, 11G16

Tsuyoshi Itoh
Affiliation: College of Science and Engineering, Ritsumeikan University, 1-1-1 Noji Higashi, Kusatsu, Shiga, 525-8577, Japan
Email: tsitoh@se.ritsumei.ac.jp

DOI: 10.1090/S0002-9939-08-09506-3
PII: S 0002-9939(08)09506-3
Received by editor(s): September 4, 2007,
Received by editor(s) in revised form: January 23, 2008
Posted: August 20, 2008
Communicated by: Ken Ono
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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