Upper bounds on class numbers of real quadratic fields

TL;DR

This paper improves bounds on the number of real quadratic fields with small class numbers, extends results to multiple discriminants, and provides algebraic conditions for fundamental units, correcting previous work.

Contribution

It offers improved upper bounds on class numbers of real quadratic fields, generalizes estimates to multiple discriminants, and introduces algebraic criteria for fundamental units, correcting prior inaccuracies.

Findings

At least x^{1/2 - ε} real quadratic fields have class numbers below a specified bound.

Established a similar estimate for m-tuples of discriminants for any m ≥ 1.

Provided algebraic conditions to lower bound the size of fundamental units.

Abstract

We prove that, for any $\varepsilon>0$, the number of real quadratic fields $\mathbb{Q}(\sqrt{d})$ of discriminant $d<x$ whose class number is $\ll \sqrt{d}(\log{d})^{-2}(\log\log{d})^{-1}$ is at least $x^{1/2-\varepsilon}$ for $x$ large enough. This improves by a factor $\log\log{d}$ a result from 1971 by Yamamoto. We also establish a similar estimate for $m$-tuples of discriminants for any $m\geq 1$. Finally, we provide algebraic conditions to give a lower bound for the size of the fundamental unit of $\mathbb{Q}(\sqrt{d})$, generalizing a criterion by Yamamoto. Our proof corrects a work of Halter-Koch.