A note on the relative class number of the cyclotomic \(\mathbb Z_p\)-extension of \(\mathbb Q(\sqrt{-p})\). (English) Zbl 1333.11103

Ichimura, Humio; Nakajima, Shoichi

Proc. Japan Acad., Ser. A 88, No. 1, 16-20 (2012).

Summary: Let \(p\) be a prime number with \(p \equiv 3 \bmod 4\) and \(q=(p-1)/2\). Let \(k=\mathbb Q(\sqrt{-p})\) and \(k_{\infty}/k\) be the cyclotomic \(\mathbb Z_p\)-extension. Denote by \(h_n^{-}\) the relative class number of the \(n\)-th layer \(k_{n}\). Let \(\ell\) be a prime number with \(\ell \neq p\). We show that, for any \(n \geq 1, \ell\) does not divide \(h_n^{-}/h_{n-1}^{-}\) (resp. \(h_1^{-}/h_0^{-}\)) if \(\ell\) is a primitive root modulo \(p^2\) (resp. \(p\)) and \(\ell \geq q-2\) (resp. \(\ell \geq q-6\)). Further, we show with the help of computer that when \(p < 10000\) and \(n \leq 100\), \(\ell\) does not divide \(h_n^{-}/h_{n-1}^{-}\) (resp. \(h_1^{-}/h_0^{-}\)) for any prime \(\ell\) which is a primitive root modulo \(p^2\) (resp. \(p\)).

MSC:

11R29	Class numbers, class groups, discriminants
11R11	Quadratic extensions
11R18	Cyclotomic extensions

Keywords:

class number; quadratic field; cyclotomic extension; non-p part

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