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Computing Iwasawa modules of real quadratic number fields. (English) Zbl 0840.11043
For a number field \(F\), the Iwasawa module means the projective limit of the \(p\)-parts of the ideal class groups in the cyclotomic \(\mathbb{Z}_p\)-extension \(F_\infty\) of \(F\). In the Abelian case, since the Iwasawa cyclotomic \(\mu\)-invariant is zero by a theorem of B. Ferrero and L. C. Washington [Ann. Math., II. Ser. 109, 377-395 (1979; Zbl 0443.12001)], the Iwasawa module is a finitely generated \(\mathbb{Z}_p\)-module. Further, in the real Abelian case (more generally totally real case), it is conjectured that the Iwasawa module is finite, i.e., both of the Iwasawa cyclotomic \(\lambda\)- and \(\mu\)-invariants are always zero. This is often called Greenberg’s conjecture [R. Greenberg, Am. J. Math. 98, 263-284 (1976; Zbl 0334.12013)].
In this paper, focusing on the case that \(F\) is a real quadratic field in which a given odd prime number \(p\) does not split, the authors give an effective method to compute the structure of the Iwasawa module of \(F\) for \(p\). As an illustration of the method, they compute the Iwasawa modules for \(p= 3\) and all real quadratic fields \(F= \mathbb{Q}(\sqrt f)\) with conductor \(f< 10000\) and \(f\not\equiv 1\pmod 3\). In the end, they verify that Greenberg’s conjecture is true for all these \(F\)’s when \(p= 3\). They also notice that their method applies to more general cases.
Their method is based on the following main result: Let \(\chi\) be the non-trivial character associated to \(k\), \(F_n\) the \(n\)th layer of \(F_\infty/F\), and \(C_n\) the \(\chi\)-part of the dual group of the unit group \(E_n\) of \(F_n\) modulo its cyclotomic unit group \(Cyc_n\); \(C_n= \text{Hom}(E_n/Cyc_n, \mathbb{Q}/\mathbb{Z})^\chi\). Put \(G_n= \text{Gal}(F_n/F)\) and \(\eta_n= N_{\mathbb{Q}(\zeta_{f_n})/F_n}(1- \zeta_{f_n})^{1- \sigma}\), where \(f_n\) is the conductor of \(F_n\), \(N_{\mathbb{Q}(\zeta_{f_n})/F_n}\) the norm map from the \(f_n\)th cyclotomic field \(\mathbb{Q}(\zeta_{f_n})\) to \(F_n\) and \(\sigma\) is the generator of the Galois group of \(F_n\) over \(\mathbb{Q}(\zeta_{p^{n+ 1}}+ \zeta^{- 1}_{p^{n+ 1}})\). Then, for any \(m> 0\) and \(n\geq 0\), there is a canonical surjective \(G_n\)-homomorphism from the finite local ring \(\mathbb{Z}/p^m \mathbb{Z}[G_n]\) to \(C_n/p^m C_n\) with kernel \(\{f(\eta_n)\mid f\in \text{Hom}_{G_n}(E_n, \mathbb{Z}/p^m\mathbb{Z}[G_n])\}\). The kernel is further given more explicitly in terms of the Frobenius elements of certain Kummer extensions. This paper also shows that if there exists \(n_0\geq 0\) such that \(C_{n_0}= C_{n_0+ 1}\), then \(C_n\) stabilizes for \(n\geq n_0\), and moreover, there is an isomorphism of \(G_n\)-modules from the \(\chi\)-part of the \(p\)-part \(A_n\) of the ideal class group of \(F_n\) to the \(\chi\)-part of \(E_n/Cyc_n\) for \(n\geq n_0\). These enable us to compute \(C_n/p^m C_n\) for various \(m\), that is, the \(\chi\)-part of \(A_n\), by calculation only in a finite group ring \(\mathbb{Z}/p^m \mathbb{Z}[T]\).
Finally, an important remark has to be made: Table 5.2 of this paper contains some mistakes, which are found by F. Fukuda, H. Ichimura, K. Komatsu, M. Ozaki, H. Sumida and the reviewer (for such other computational data, see the paper of R. Greenberg cited above, the paper of A. Candiotti [Compos. Math. 29, 89-111 (1974; Zbl 0364.12003)], the paper of K. Fukuda and K. Komatsu [Math. Comput. 65, 313-318 (1996)] and the paper of H. Ichimura and H. Sumida entitled “On the Iwasawa \(\lambda\)-invariants of certain real Abelian fields” [to appear in Tohoku Math. J.]). These are the cases \(f= 476, 1016, 5081, 5297\) and 6584 (the first two cases never even appear in Table 5.2). According to the letter from the second author to the reviewer, the correct data of the projective limit \(C\) of \(C_n\) for these \(f\)’s are as follows: \(C= \mathbb{Z}/ 243\mathbb{Z}\), \(\mathbb{Z}/ 243\mathbb{Z}\), \(\mathbb{Z}/81\mathbb{Z}\), \(\mathbb{Z}/81\mathbb{Z}\) and \(\mathbb{Z}/243\mathbb{Z}\) (Greenberg’s conjecture is true) for \(f= 476, 1016, 5081, 5297\) and 6584, respectively. For more details, see an erratum which the authors will write in the near future.
Reviewer: H.Taya (Tokyo)

MSC:
11R23 Iwasawa theory
11R11 Quadratic extensions
11Y40 Algebraic number theory computations
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References:
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