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 $4+r+3s\le 3^{n+1}$. 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 $\mathbb{Q}(\sqrt{f_1(D)}),\dots ,\mathbb{Q}(\sqrt{f_r(D)})$ and $\mathbb{Q}(\sqrt{-f_{r+1}(D)}),\dots ,\mathbb{Q}(\sqrt{-f_{r+s}(D)})$ 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 $\mathbb{Q}(\sqrt{a+g_1(d)}),\dots ,\mathbb{Q}(\sqrt{a+g_k(d)})$ is simultaneously less than $n$ for polynomials $g_1(x),g_2(x),\dots ,g_k(x)$ that take integer values at the integers and have no constant terms.