Consecutive pure fields of the form $\mathbb{Q}\left(\sqrt[l]{a}\right)$ with large class numbers

TL;DR

The paper studies the growth of class numbers in consecutive pure $l$-th root fields $\mathbb{Q}(\sqrt[l]{d+j})$ by linking $h_K$ to special values of automorphic $L$-functions. It develops a short Dirichlet-sum approximation for $\log L(1,\tilde{\pi}_{d_K})$ under Langlands conjecture and uses a sieve plus residue-symbol control to force favorable splitting of primes in the discriminants $D_j$; these ingredients yield a lower bound $x^{1/l-\mathrm{o}(1)}$ for the number of suitable $d\le x$. A regulator bound (Hypothesis 1) is invoked to convert $L$-value growth into unbounded class numbers for all $k$ consecutive pure fields, thereby obtaining arbitrarily large $h$ in the family. The work extends prior results for $l=3$ to general prime $l$ by combining analytic, algebraic, and sieve methods and highlights the role of regulator behavior in achieving large class numbers in families of higher-degree fields.

Abstract

Let $l$ be a rational prime greater than or equal to $3$ and $k$ be a given positive integer. Under a conjecture due to Langland and an assumption on upper bound for the regulator of fields of the form $\mathbb{Q}\left(\sqrt[l]a\right)$, we prove that there are atleast $x^{1/l-o(1)} $ integers $1\leq d\leq x$ such that the consecutive pure fields of the form $\mathbb{Q}\left(\sqrt[l]{d+1}\right), \dots ,\mathbb{Q}\left(\sqrt[l]{d+k}\right) $ have arbitrary large class numbers.

Theorems & Definitions (14)

• Theorem 1: DBDY
• Theorem 2
• Remark 1
• Proposition 2.1
• proof
• Remark 2
• Proposition 2.2: Landau
• Lemma 3.1
• proof
• Lemma 3.2