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

Abstract

Let $n\ge1$, $r\ge0$ and $s\ge0$ be integers satisfying $4+r+3 s\le3^{n+1}$. 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 $n$. Moreover, for any positive integer $k$, there exist positive integers $a, d$ such that the 3-rank of the class group of each quadratic fields $\mathbb{Q}(\sqrt{a+g_1(d)}), \ldots,\mathbb{Q}(\sqrt{a+g_k(d)})$ is simultaneously less than $n$ for polynomials $g_1(x), g_2(x), \ldots, g_k(x)$ that take integer values at the integers and have no constant terms.