Densities for 4-class ranks of totally complex quadratic extensions of real quadratic fields (Q1900883)

Let \(K\) be a quadratic extension of a real quadratic field \(F= \mathbb{Q} (\sqrt {d})\) where \(d\) is divisible by \(k+1\) primes, none of which are congruent to 3 modulo 4. Furthermore, assume the fundamental unit of \(F\) has negative norm and the 2-class group is elementary. Let \(R_K\) denote the 4-rank of the class group of \(K\), \({\mathcal D}_{K/F}\) the relative discriminant of \(K/F\) and \(N({\mathcal D}_{K/F})\) the absolute norm of \({\mathcal D}_{K/F}\). For positive integers \(t\), nonnegative integers \(j\) and positive real numbers \(x\), define \(A_t =\) \{totally complex quadratic extensions \(K\) of \(F\) with exactly \(t\) finite primes of \(F\) ramifying in \(K\)\}, \(A_{t_j x}= \{K\in A_t: N({\mathcal D}_{K/F} )\leq x\}\), \(A_{t,j;x} = \{K \in A_{t,x}: R_K= j\}\), \[ d_{t,j} = \lim_{x\to \infty} {{|A_{t,j; x}|} \over {|A_{t,x}|}}, \qquad d_{\infty,j}= \lim_{t\to \infty} d_{t,j}. \] The author proves that \[ d_{\infty,j}= \begin{cases} 0 &\quad \text{if }j< k,\\ {{2^{-(j- k)^2} \prod^\infty_{i=1} (1- 2^{-i})} \over {\prod^{j- k}_{i= 1} (1- 2^{-i} )^2}} &\quad \text{if }j\geq k. \end{cases} \] To obtain these results, it is shown that for \(K\in A_t\), rank \(C_K= t-1\), where \(C_K\) denotes the 2-class group of \(K\), and that \(R_K= k+ t+ 1- \text{rank } M_K\) where \(M_K\) is a \((t+1) \times (t+2)\) matrix over the field of two elements.