On infinite unramified Galois extensions of algebraic number fields with many primes decomposing almost completely (Q1086620)

Let \(M/K\) be an infinite unramified Galois extension of number fields. Let \(S_0\) denote the set of all nonarchimedean primes of \(K\), \(f(P)\) the residue extension degree of \(P\in S_0\) in \(M/K\), \(N(P)\) the absolute norm of \(P\in S_0\), \(S\) the subset \(\{P\in S_0\mid f(P)<\infty \}\) and \(S_{\infty}\) the set of all archimedean primes of \(K\). Define the constant \(\alpha_P\), for each prime \(P\in S\) by \(\alpha_P=(N(P)^{f(P)/2}-1)^{-1} \text{Log}\, N(P)\) and for \(P\in S_{\infty}\) real (resp. imaginary) by \(\alpha_P=\tfrac12 (\text{Log}\, 8\pi +\gamma +\tfrac{\pi}{2})\) (resp. \(\alpha_ P=\text{Log}\, 8\pi +\gamma)\), \(\gamma\) being Euler's constant. Under the generalized Riemann hypothesis, \textit{Y. Ihara} proved the inequality \[ \sum_{P\in S\cup S_{\infty}}\alpha_P\leq \tfrac12 \text{Log}\, | d_K| \] where \(d_K\) is the discriminant of \(K\) [cf. ibid. 35, 693--709 (1983; Zbl 0518.12006)]. Moreover, for the ratio \[ \rho (M/K)=\bigl(\tfrac12 \text{Log}\, | d_K|\bigr)^{-1}\bigl(\sum_P \alpha_P\bigr), \] he gave an example with \(\rho(M/K)\geq 0.7517\dots\) and then asked for extensions \(M/K\) such that \(\rho(M/K)=1\) (loc. cit.). In the paper under review, the author gives a way to construct examples of \(M/K\) with large \(\rho(M/K)\). Combining Ihara's work, Golod-Shafarevich theory, and a result of \textit{J. Martinet} on class field towers [Invent. Math. 44, 65--73 (1978; Zbl 0369.12007)], he gets an extension \(M/K\) with \(\rho(M/K)\geq 0.9115\dots\).