Algorithmic approach to logarithmic class groups
The authors describe an efficient algorithm for computing the logarithmic class group of an algebraic number field \(K\) in connection with the wild kernel of \(K_2(k)\). Logarithmic classes were introduced by the reviewer [\textit{J. F. Jaulent}, J. Théor. Nombres Bordx. 6, 307-327 (1994; Zbl 0827.11064)] and so-called because their definition involves \(\ell\)-adic valuations using the Iwasawa logarithm of local norms. By \(\ell\)-adic class field theory the logarithmic \(\ell\)-class group \(\widetilde{Cl}_k\) correspond to the Galois group \(\text{Gal}(K^{lc}/K^c)\) over the cyclotomic \(\mathbb{Z}_\ell\)-extension \(K^c\) of the maximal abelian pro-\(\ell\)-extension \(K^k\) of \(K\) which is locally trivial over \(K^c\). But its main interest is the canonical isomorphism \[ \mu_{\ell}\otimes\widetilde{Cl}_K\simeq H_2(K)/H_2(K)^\ell \] given by the reviewer [Acta Arith. 67, 335-348 (1994; Zbl 0835.11042)] where \(H_2(K)\) is the Hilbert kernel in \(K_2(K)\) and \(K\) is assumed to contain the \(2\ell\)-th roots of unity. In this paper the authors describe from an algorithmic point of view the structure of the logarithmic \(\ell\)-class group \(\overline{Cl}_K\) for finite Galois extensions \(K\) of \(\mathbb{Q}\), and they illustrate their method using the PARI package by performing the computation of \(\widetilde{Cl}_K\) for biquadratic number fields \(\mathbb{Q}[\sqrt{-1},\sqrt d]\) in the case \(\ell=2\) and \(\mathbb{Q}[\sqrt{-3},\sqrt d]\) in the case \(\ell=3\) for \(d\leq 2000\). As explained above, their results give also the \(\ell\)-rank of the Hilbert kernel for these fields.
scientific article; zbMATH DE number 1310080
scientific article