On the 2-part of the ideal class group of the cyclotomic \(\mathbb Z_p\)-extension over the rationals
Let \(p\not=l\) be two primes with \(p>2\). For a number field \(F\), let \(F_{\infty}/F\) the cyclotomic \(\mathbb{Z}_p\)-extension and \(F_n\) the \(n\)-th layer of \(F_{\infty}/F\). Moreover, we denote by \(h_{F_n}\) the class number of \(F_n\), and by \(Cl_{F_{\infty}}\) the direct limit of the ideal class groups of \(F_n\). It is a classical result of \textit{L. C. Washington} [Invent. Math. 49, 87--97 (1978; Zbl 0403.12007)] that the quotient \(h_{F_n} / h_{F_{n-1}}\) is not divisible by \(l\) for \(n > N(p)\) for sufficiently large \(N(p)\), i.e. the \(l\)-part \(Cl_{F_{\infty}}(l)\) of \(Cl_{F_{\infty}}\) is a finite abelian group. In the paper under review the authors show that \(Cl_{F_{\infty}}(2)\) is trivial if \(p < 500\) and either \(F = \mathbb{Q}\) or the order of \(2 \bmod p\) is even and \(F\) is the maximal subfield of the \(p\)-th cyclotomic field \(\mathbb{Q}(\zeta_p)\) such that \([F:\mathbb{Q}]\) is a power of two. For this, they use an expicit bound \(N(p)\) as above which is easily obtained from work of \textit{K. Horie} [J. Lond. Math. Soc., II. Ser. 66, No. 2, 257--275 (2002; Zbl 1011.11072); Tôhoku Math. J. (2) 61, No. 4, 551--570 (2009; Zbl 1238.11101)].
scientific article; zbMATH DE number 5821521
scientific article