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A note on 4-rank densities. (English) Zbl 1127.11076

The author considers densities for the \(4\)-ranks of the tame kernels \(K_2(\mathcal O_F)\), where \(F\) runs through the quadratic number fields \(\mathbb Q(\sqrt{p_1p_2p_3})\) with \(p_1, p_2, p_3\) distinct primes \(\equiv 1 \bmod 8\). The main result shows that the \(4\)-ranks \(0,1,2,\) and \(3\) appear with natural densities \(\frac{1}{4}, \frac{17}{32}, \frac{13}{64}\) and \(\frac{1}{64}\), respectively.
The proof is based on the \(4\)-rank description of the tame kernels of quadratic number fields via “Rédéi”-matrices in [J. Hurrelbrink and M. Kolster, “Tame kernels under relative quadratic extensions and Hilbert symbols”, J. Reine Angew. Math. 499, 145–188 (1998; Zbl 1044.11100)], and the general approach towards density results in this situation described in [R. Osburn, “Densities of 4-ranks of \(K_2({\mathcal O})\)”, Acta Arith. 102, No. 1, 45–54 (2002; Zbl 1029.11062)].

MSC:

11R70 \(K\)-theory of global fields
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
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