On Greenberg’s conjecture on a certain real quadratic field. (English) Zbl 0990.11065

Mizusawa, Yasushi

Proc. Japan Acad., Ser. A 76, No. 10, 163-164 (2000).

Let \(p\) be a prime number and let \(F_{\infty} = \cup_{n \geq 0} F_n\) be the cyclotomic \(\mathbb Z _p\)-extension of a totally real number field \(F\). A conjecture of R. Greenberg’s [Am. J. Math. 98, 263-284 (1976; Zbl 0334.12013)] predicts that the power of \(p\) dividing the class number of \(F_n\) is bounded independently of \(n\). Here, using a criterion of H. Ichimura and H. Sumida [Tôhoku Math. J. (2) 49, No. 2, 203-215 (1997; Zbl 0886.11060)], the author shows that this conjecture holds in the case of \(p = 3\) and the quadratic number field \(\mathbb{Q}(\sqrt{39345017})\), which has an infinite \(3\)-class field tower.

Reviewer: Abbas Movahhedi (Limoges)

MSC:

11R23	Iwasawa theory
11R11	Quadratic extensions

Keywords:

Iwasawa invariants; class field towers; real quadratic field

Citations:

Zbl 0334.12013; Zbl 0886.11060

References:

[1]	Ichimura, H., and Sumida, H.: On the Iwasawa invariants of certain real abelian fields. Tohoku Math. J., 49 , 203-215 (1997). · Zbl 0886.11060
[2]	Ichimura, H., and Sumida, H.: On the Iwasawa invariants of certain real abelian fields II. Inter. J. Math., 7 , 721-744 (1996). · Zbl 0881.11075
[3]	Ozaki, M.: Iwasawa invariants of \(p\)-extensions of totally real number fields (preprint). · Zbl 1253.11100
[4]	Schoof, R.: Infinite class field towers of quadratic fields. J. Reine Angew. Math., 372 , 209-220 (1986). · Zbl 0589.12011
[5]	Washington, L.: Introduction to Cyclotomic Fields. Grad. Texts in Math., vol. 83, Springer, New York (1982). · Zbl 0484.12001