An application of Birch-Tate formula to tame kernels of real quadratic number fields (Q6617373)

Let \(K\) be a real quadratic field, \(\mathbb Z_K\) its ring of integers and \(K_2(\mathbb Z_K)\) the tame kernel of \(K\). \textit{B. J. Birch} [Proc. Symp. Pure Math. 20, 87--95 (1971; Zbl 0218.12010)] and \textit{J. Tate} [in: Actes Congr. Int. Math., Nice 1970, Tome 1, 201--211 (1971; Zbl 0229.12013)] conjectured a formula for the cardinality of \(K_2(\mathbb Z_K)\), which has been later established for all real abelian fields (see the footnote on p. 499 in the paper by \textit{A. Wiles} [Ann. Math. (2) 131, No. 3, 493--540 (1990; Zbl 0719.11071)] and in the case of real quadratic \(K\) gives\N\[\N\#K_2(\mathbb Z_K)=-2L(\chi,-1),\N\]\Nwhere \(\chi\) is the Dirichlet quadratic character associated to \(K\) .\N\NThe authors use this formula to determine the \(2\)-part \(K_2(\mathbb Z_K)_2\) of \(K_2(\mathbb Z_K)\) in certain cases. In Theorem 2 they show that if \(K=\mathbb Q(\sqrt D)\) with \(D=4p_1\cdots p_n\), where \(n\) is even, \(p_i\) are distinct primes with \(p_1\equiv3\) mod \(8\), \(p_i\equiv5\) mod \(8\) for \(i=2,3,\dots,n\) and\N\[\N\left(\frac{p_i}{p_j}\right)=-1\ \text{for\ } 2\le i<j\le n,\N\]\N\N\[\N\left(\frac{p_1}{p_2}\right)=-1\ \text{and\ } \left(\frac{p_1}{p_i}\right)=1\ \text{for\ } i\ge3,\N\]\Nthen\N\[\NK_2(\mathbb Z_K)_2=C_2^{n-1}\times C_2^\delta\N\]\Nwith some \(\delta\ge3\) . A similar result is obtained in part (2) of Theorem 3 for the case of odd \(n\). Part (1) of Theorem 3 shows that if \(n\ge3\) is odd, \(D=p_1\cdots p_n\) are distinct primes congruent to \(5\) mod \(8\) and for \(1\le i<j\le n\) one has\N\[\N\left(\frac{p_i}{p_j}\right)=-1,\N\]\Nthen\N\[\NK_2(\mathbb Z_K)_2=C_2^{n+1}.\N\]\NIn the case \(n=1\) one has\N\[\NK_2(\mathbb Z_K)_2=C_2^3.\N\]\N\NThe authors note that Theorem 3 generalizes a result of \textit{J. Browkin} and \textit{A. Schinzel} [J. Reine Angew. Math. 331, 104--113 (1982; Zbl 0493.12013)] whose Corollary 1 deals with the case \(n=1\).