An asymptotic on the logarithms of the relative class numbers of imaginary abelian number fields of prime conductor and large degree

TL;DR

The paper analyzes relative class numbers $h_{\mathbb K}^-$ of imaginary abelian number fields of prime conductor $p$ and large degree $m=(p-1)/d$, establishing an explicit asymptotic for $\log h_{\mathbb K}^-$ under the condition $\phi(d)=o(\log p)$. Central to the approach is the relative class number formula and the mean-square analysis of $L(1,\chi)$ values over the odd Dirichlet characters $\chi$ in $X_p^-(H)$, with detailed bounds on the auxiliary function $f_H(s)=\sum_{\chi\in X_p^-(H)}\log L(s,\chi)$. The authors derive explicit bounds for $|f_H(1)|$ and $|f_H(\sigma)|$ (for $\sigma>1$) by decomposing Dirichlet series into ranges and applying Montgomery–Vaughan-type estimates and Borel–Carathéodory arguments, which together yield $h_{\mathbb K}^- = w_{\mathbb K}\left(\frac{p}{4\pi^2}\right)^{m/4}\exp(o(1))$. They also show the asymptotic fails under a slightly weaker restriction $\phi(d)=O(\log p)$, via a Mersenne-prime based construction, and provide an alternative, shorter proof under a weaker assumption via Montgomery–GS1, highlighting the result’s dependence on growth conditions for $d$.

Abstract

An asymptotic on the logarithms of the relative class numbers of the cyclotomic number fields of prime conductors $p$ is known. Here we give an asymptotic on the logarithms of the relative class numbers of the imaginary abelian number fields of prime conductors $p$ and large degrees $m =(p-1)/d$ with $φ(d)=o(\log p)$. We also show that this asymptotic does not hold true anymore under the only slightly weaker restriction $φ(d)=O(\log p)$.

Theorems & Definitions (12)

• Proposition 1
• Proposition 2
• Theorem 3
• Theorem 4
• Lemma 5
• Lemma 6
• Lemma 7
• proof
• Lemma 8
• proof