On the 3-rank of the class group of quadratic fields
Shi-Chao Chen, Chuan-Chuan Wu
TL;DR
The paper tackles the problem of bounding the 3-rank of class groups across families of quadratic fields of the form Q(√{f(D)}) with D ranging over fundamental discriminants. It develops a density-based framework combining Nakagawa–Horie refinements of Davenport– Heilbronn with a polynomial-extension argument (Bergelson–Leibman) to establish simultaneous bounds r_3 on multiple quadratic fields defined by linear and polynomial families. The main contributions include Theorem th1, which yields infinite sets of D for which r_3(<n) holds simultaneously for all prescribed linear forms (and their negatives), and Theorem density, which ensures the existence of a,d producing r_3(a+g_i(d))<n for several polynomials g_i with no constant term. Collectively, these results extend the study of 3-torsion in class groups to broader families of quadratic fields and illuminate the density structure underpinning simultaneous small 3-ranks.
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.