On the 3-rank of the class group of quadratic fields

Abstract Let $n\ge 1$ , $r\ge 0$ , and $s\ge 0$ be integers satisfying $4+r+3 s\le 3^{n+1}$ . Given linear polynomials $f_{i}(x)=m_{i} x+n_{i}$ for $1 \le i \le r+s$ , where the coefficients $m_{i} , n_{i}$ are positive integers satisfying certain conditions, we prove that there exist infinitely many fundamental discriminants $D>0$ such that the 3-rank of the class group of each quadratic fields $\mathbb {Q}(\sqrt {f_1(D)}), \ldots , \mathbb {Q}(\sqrt {f_r(D)})$ and $\mathbb {Q}(\sqrt {-f_{r+1}(D)}), \ldots , \mathbb {Q}(\sqrt {-f_{r+s}(D)})$ is simultaneously less than n . For a positive integer $k $ , let $g_1,\dots ,g_k\in \mathbb {Q}[x]$ be polynomials taking integer values at integers with $g_i(0)=0$ . We also prove that there exist positive integers $a,d$ , such that each $a+g_i(d)$ is a fundamental discriminant and the 3-rank of the class group of each quadratic field $\mathbb {Q}(\sqrt {a+g_1(d)}), \ldots ,\mathbb {Q}(\sqrt {a+g_k(d)})$ is simultaneously less than n . Moreover, these discriminants can be chosen all positive or all negative, giving either all real or all imaginary quadratic fields $\mathbb {Q}(\sqrt {a+g_i(d)})$ .