Large moments and extreme values of class numbers of indefinite binary quadratic forms

TL;DR

This paper derives an asymptotic formula for the moments of class numbers of indefinite binary quadratic forms, explores the distribution of large class numbers, and provides bounds related to fundamental units and class numbers.

Contribution

It extends previous work by providing a uniform asymptotic for the moments of class numbers over a broad range of parameters and analyzes the distribution of large class numbers in relation to fundamental units.

Findings

Asymptotic formula for the $k$-th moment of $h(d)$ for a wide range of $k$.

Distribution of large class numbers matches that of imaginary quadratic fields.

Established lower bounds for class numbers in terms of fundamental units.

Abstract

Let $h(d)$ be the class number of indefinite binary quadratic forms of discriminant $d$, and let $\varepsilon_d$ be the corresponding fundamental unit. In this paper, we obtain an asymptotic formula for the $k$-th moment of $h(d)$ over positive discriminants $d$ with $\varepsilon_d\leq x$, uniformly for real numbers $k$ in the range $0<k\leq (\log x)^{1-o(1)}$. This improves upon the work of Raulf, who obtained such an asymptotic for a fixed positive integer $k$. We also investigate the distribution of large values of $h(d)$ when the $d$'s are ordered according to the size of their fundamental units $\varepsilon_d$. In particular, we show that the tail of this distribution has the same shape as that of class numbers of imaginary quadratic fields ordered by the size of their discriminants. As an application of these results, we prove that there are many positive discriminants $d$ with class number $h(d)\geq (e^{\gamma}/3+o(1))\cdot\varepsilon_d (\log\log \varepsilon_d)/\log \varepsilon_d$, a bound that we believe is best possible. We also obtain an upper bound for $h(d)$ that is twice as large, assuming the generalized Riemann hypothesis.