Transactions of the American Mathematical Society

Author(s): Jangheon Oh
Journal: Trans. Amer. Math. Soc. 350 (1998), 3639-3655.
MSC (1991): Primary 11R23
Retrieve article in: PDF
This article is available free of charge

Abstract: We study the relation between zeta-functions and Iwasawa modules. We prove that the Iwasawa modules $X^{-}_{k({\zeta }_{p})}$ for almost all

determine the zeta function ${\zeta }_{k}$ when

is a totally real field. Conversely, we prove that the $\lambda$ -part of the Iwasawa module $X_{k}$ is determined by its zeta-function ${\zeta }_{k}$ up to pseudo-isomorphism for any number field

Moreover, we prove that arithmetically equivalent CM fields have also the same ${\mu }^{-}$ -invariant.

1.: N.Adachi and K.Komatsu, The Maximal $p$ -extensions and Zeta-Functions of Algebraic Number Fields, Memoirs of the School of Science & Engineering Waseda Univ. 51 (1987), 25-31. MR 90a:11135
2.: B. Ferrero and L.Washington, The Iwasawa invariant ${\mu }_{p}$ vanishes for abelian number fields, Ann. of Math. 109 (1979), 377-395. MR 81a:12005
3.: F.Gassmann, Bererkungen zu der vorstehenden arbeit von Hurwitz, Math.Z. 25 (1926), 124-143.
4.: D.Goss and W.Sinnott, Special Values of Artin $L$ -series, Math.Ann. 275 (1986), 529-537. MR 87k:11127
5.: K.Iwasawa, On $p$ -adic $L$ -functions, Ann. of Math. 89 (1969), 198-205. MR 42:4522
6.: K.Iwasawa, On ${\mathbb{Z}}_{\ell }$ -extensions of algebraic number fields, Ann. of Math. 98 (1973), 246-326. MR 15:509d
7.: K.Iwasawa, On the $\mu$ -invariants of ${\mathbb{Z}}_{\ell }$ -extensions, Number Theory, Algebraic Geometry and Commutative Algebra (in honor of Y. Akizuki), Kinokuniya, Tokyo, 1973, pp. 1-11. MR 50:9839
8.: K.Komatsu, On Zeta-functions and cyclotomic ${\mathbb{Z}}_{p}$ -extensions of algebraic number fields, Tôhoku Math. Journ. 36 (1984), 555-562. MR 86a:11046
9.: R.Perlis, On the equation ${\zeta }_{k} = {\zeta }_{k'}$ , J. Number Theory 9 (1977), 342-360. MR 56:5503
10.: R.Perlis, On the class numbers of arithmetically equivalent fields, J. Number Theory 10 (1978), 489-509. MR 80c:12014
11.: R.Perlis and N.Colwell, Iwasawa Invariants and Arithmetic Equivalence, unpublished.
12.: J.Tate, Endomorphism of abelian varieties over finite fields, Invent. Math. 2 (1966), 134-144. MR 34:5829
13.: S.Turner, Adele rings of global field of positive characteristic, Bol. Soc. Brasil. Math. 9 (1978), 89-95. MR 80c:12017
14.: L.Washington, Introduction to Cyclotomic Fields, Springer, Berlin, Heidelberg, New York, 1982. MR 85g:11001
15.: A.Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. 131 (1990), 493-540. MR 91i:11163

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11R23

Jangheon Oh
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Address at time of publication: KIAS, 207-43 Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-012, Korea
Email: ohj@kias.kaist.ac.kr

DOI: 10.1090/S0002-9947-98-01967-9
PII: S 0002-9947(98)01967-9
Keywords: Iwasawa module, zeta function, $p$-adic $L$-function
Received by editor(s): April 16, 1996
Received by editor(s) in revised form: June 7, 1996 and October 23, 1996
Additional Notes: This paper is part of the author's Ph.D thesis. I would like to thank my adviser, W. Sinnott, for introducing me to this subject, for pointing out to me the key idea and for many valuable comments
Copyright of article: Copyright 1998, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS