On the vanishing of Iwasawa invariants of absolutely abelian p-extensions

Gen Yamamoto

Gen Yamamoto

Acta Arithmetica, Tome 92 (2000), p. 365-371 / Harvested from The Polish Digital Mathematics Library

Résumé

1. Introduction. Let p be a prime number and $ℤ_{p}$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty}$ a $ℤ_{p}$ -extension of k, $k_{n}$ the nth layer of $k_{\infty} / k$ , and $A_{n}$ the p-Sylow subgroup of the ideal class group of $k_{n}$ . Iwasawa proved the following well-known theorem about the order $A_{n}$ of $A_{n}$ : Theorem A (Iwasawa). Let $k_{\infty} / k$ be a $ℤ_{p}$ -extension and $A_{n}$ the p-Sylow subgroup of the ideal class group of $k_{n}$ , where $k_{n}$ is the $n$ th layer of $k_{\infty} / k$ . Then there exist integers $λ = λ (k_{\infty} / k) \geq 0$ , $μ = μ (k_{\infty} / k) \geq 0$ , $ν = ν (k_{\infty} / k)$ , and n₀ ≥ 0 such that $A_{n} = p^{λ n + μ p^{n} + ν}$ for all n ≥ n₀, where $A_{n}$ is the order of $A_{n}$ . These integers $λ = λ (k_{\infty} / k)$ , $μ = μ (k_{\infty} / k)$ and $ν = ν (k_{\infty} / k)$ are called Iwasawa invariants of $k_{\infty} / k$ for p. If $k_{\infty}$ is the cyclotomic $ℤ_{p}$ -extension of k, then we denote λ (resp. μ and ν) by $λ_{p} (k)$ (resp. $μ_{p} (k)$ and $ν_{p} (k)$ ). Ferrero and Washington proved $μ_{p} (k) = 0$ for any abelian extension field k of ℚ. On the other hand, Greenberg [4] conjectured that if k is a totally real, then $λ_{p} (k) = μ_{p} (k) = 0$ . We call this conjecture Greenberg’s conjecture. In this paper, we determine all absolutely abelian p-extensions k with $λ_{p} (k) = μ_{p} (k) = ν_{p} (k) = 0$ for an odd prime p, by using the results of G. Cornell and M. Rosen [1].

Publié le : 2000-01-01

Zbl 0964.11048

EUDML-ID : urn:eudml:doc:207435

@article{bwmeta1.element.bwnjournal-article-aav94i4p365bwm,
     author = {Gen Yamamoto},
     title = {On the vanishing of Iwasawa invariants of absolutely abelian p-extensions},
     journal = {Acta Arithmetica},
     volume = {92},
     year = {2000},
     pages = {365-371},
     zbl = {0964.11048},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav94i4p365bwm}
}

Gen Yamamoto. On the vanishing of Iwasawa invariants of absolutely abelian p-extensions. Acta Arithmetica, Tome 92 (2000) pp. 365-371. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav94i4p365bwm/

Bibliographie

[00000] [1] G. Cornell and M. Rosen, The class group of an absolutely abelian l-extension, Illinois J. Math. 32 (1988), 453-461. | Zbl 0654.12006

[00001] [2] T. Fukuda, On the vanishing of Iwasawa invariants of certain cyclic extensions of ℚ with prime degree, Proc. Japan Acad. 73 (1997), 108-110. | Zbl 0899.11052

[00002] [3] T. Fukuda, K. Komatsu, M. Ozaki and H. Taya, On Iwasawa $λ_{p}$ -invariants of relative real cyclic extension of degree p, Tokyo J. Math. 20 (1997), 475-480. | Zbl 0919.11068

[00003] [4] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263-284. | Zbl 0334.12013

[00004] [5] K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257-258. | Zbl 0074.03002

[00005] [6] G. Yamamoto, On the vanishing of Iwasawa invariants of certain (p,p)-extensions of ℚ, Proc. Japan Acad. 73A (1997), 45-47. | Zbl 0879.11061