On the class numbers of the \(n\)-th layers in the cyclotomic \(\mathbb{Z}_2\)-extension of \(\mathbb{Q}(\sqrt{5})\) (Q6612912)

This paper deals with problems in the context of Weber's class number problem, which is the following conjecture (or question?): All layers \(\mathbb B_n\) in the 2-cyclotomic tower over the base field \(\mathbb Q\) have class number 1. There are related questions in cases where the base field is not \(\mathbb Q\). The following is a (very likely incomplete) list of authors who have contributed to this topic: Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Sumida, Washington. We cannot give the full picture of existing results here. -- The paper under review is concerned with the 2-cyclotomic extension of \(K = \mathbb Q(\sqrt5)\). The author had previously proved [Tokyo J. Math. 39, No. 1, 69--81 (2016; Zbl 1406.11105)] that no prime less than 60000 divides the class number of any layer \(K\mathbb B_n\), and in a more recent paper [Funct. Approximatio, Comment. Math. 62, No. 1, 87--94 (2020; Zbl 1456.11215)] he had proved that for \(n=5\) (and consequently also for \(n=1,2,3,4\)) the class number is 1. The main result of the present paper says that if a prime \(\ell\) divides any class number \(h(K\mathbb B_n)\), then it must be congruent to \(\pm1\) modulo 8. (It is well known in this case that all \(h(K\mathbb B_n)\) are odd.) The proof divides neatly into two major steps of quite different flavor. First, one lets \(F_n\) be the field ``between'' \(\mathbb B_n\) and \(K\mathbb B_{n-1}\) (so \(F_n/\mathbb Q\) is a cyclic extension of degree \(2^n\)), and proves: The relative class number \(h(F_n)/h(\mathbb B_{n-1})\) is divisible by \(\ell\) iff a certain power of a specific cyclotomic unit is an \(\ell\)-th power of some element \(\eta\in F_n\); of course if \(\eta\) exists, it is again a unit. This generalizes a result known as ``Horie's Lemma''. In the second step, the technique of Mahler measures is used to prove that if the claimed congruence property of \(\ell\) is violated (that is, if \(\ell\equiv\pm3\) mod 8), the root \(\eta\) cannot exist for \(\ell>341\). Combining this lower bound with earlier results suffices to prove the main theorem. Many concepts occurring in this paper seem to have been used in earlier papers (by Aoki, and by many others), but the structure of the argument taken as a whole looks new.