On zeta functions and Iwasawa modules
HTML articles powered by AMS MathViewer
- by Jangheon Oh
- Trans. Amer. Math. Soc. 350 (1998), 3639-3655
- DOI: https://doi.org/10.1090/S0002-9947-98-01967-9
- PDF | Request permission
Abstract:
We study the relation between zeta-functions and Iwasawa modules. We prove that the Iwasawa modules $X^{-}_{k({\zeta }_{p})}$ for almost all $p$ determine the zeta function ${\zeta }_{k}$ when $k$ 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 $k.$ Moreover, we prove that arithmetically equivalent CM fields have also the same ${\mu }^{-}$-invariant.References
- Norio Adachi and Keiichi Komatsu, The maximal $p$-extensions and zeta-functions of algebraic number fields, Mem. School Sci. Engrg. Waseda Univ. 51 (1987), 25β31 (1988). MR 954760
- Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no.Β 2, 377β395. MR 528968, DOI 10.2307/1971116
- F.Gassmann, Bererkungen zu der vorstehenden arbeit von Hurwitz, Math.Z. 25 (1926), 124β143.
- David Goss and Warren Sinnott, Special values of Artin $L$-series, Math. Ann. 275 (1986), no.Β 4, 529β537. MR 859327, DOI 10.1007/BF01459134
- Kenkichi Iwasawa, On $p$-adic $L$-functions, Ann. of Math. (2) 89 (1969), 198β205. MR 269627, DOI 10.2307/1970817
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82β96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- Kenkichi Iwasawa, On the $\mu$-invariants of $Z_{\ell }$-extensions, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp.Β 1β11. MR 0357371
- Keiichi Komatsu, On zeta-functions and cyclotomic $\textbf {Z}_p$-extensions of algebraic number fields, Tohoku Math. J. (2) 36 (1984), no.Β 4, 555β562. MR 767404, DOI 10.2748/tmj/1178228762
- Robert Perlis, On the equation $\zeta _{K}(s)=\zeta _{Kβ}(s)$, J. Number Theory 9 (1977), no.Β 3, 342β360. MR 447188, DOI 10.1016/0022-314X(77)90070-1
- Robert Perlis, On the class numbers of arithmetically equivalent fields, J. Number Theory 10 (1978), no.Β 4, 489β509. MR 515057, DOI 10.1016/0022-314X(78)90020-3
- R.Perlis and N.Colwell, Iwasawa Invariants and Arithmetic Equivalence, unpublished.
- John Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134β144. MR 206004, DOI 10.1007/BF01404549
- Stuart Turner, Adele rings of global field of positive characteristic, Bol. Soc. Brasil. Mat. 9 (1978), no.Β 1, 89β95. MR 516311, DOI 10.1007/BF02584796
- Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
- A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no.Β 3, 493β540. MR 1053488, DOI 10.2307/1971468
Bibliographic Information
- 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
- 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 authorβs Ph.D thesis. I would like to thank my adviser, W.Β Sinnott, for introducing me to this subject, for pointing out to me the key idea and for many valuable comments
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 3639-3655
- MSC (1991): Primary 11R23
- DOI: https://doi.org/10.1090/S0002-9947-98-01967-9
- MathSciNet review: 1422616