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

Assuming the Langlands conjecture and a bound on regulators, this paper proves that for any prime $l \geq 3$ and integer $k$ , there exist at least $x^{1/l-o(1)}$ integers $d \leq x$ such that the consecutive pure fields $\mathbb{Q}(\sqrt[l]{d+1}), \dots, \mathbb{Q}(\sqrt[l]{d+k})$ all possess arbitrarily large class numbers.

Original authors: Jishu Das, Srilakshmi Krishnamoorthy

Published 2026-06-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jishu Das, Srilakshmi Krishnamoorthy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a mathematician exploring a vast, infinite landscape made of numbers. In this landscape, there are special "neighborhoods" called number fields. Think of a number field as a unique community where the rules of arithmetic are slightly different from the ones we use every day.

Every neighborhood has a hidden property called its class number. You can think of the class number as a measure of how "messy" or "complicated" the neighborhood is.

A low class number (like 1) means the neighborhood is tidy and orderly.
A high class number means the neighborhood is chaotic, with many different ways to organize things that don't quite fit together perfectly.

For a long time, mathematicians have been trying to find neighborhoods with arbitrarily large class numbers—places that are incredibly complex. They knew how to find these messy neighborhoods if they looked at them one by one. But a harder question remained: Can we find a whole row of consecutive neighborhoods that are all messy at the same time?

The Big Discovery

This paper, written by Jishu Das and Srilakshmi Krishnamoorthy, says yes, we can.

They focus on a specific type of neighborhood called a "pure field." Imagine these as houses built using a specific mathematical recipe involving a root (like a square root, cube root, or a 5th root). The authors prove that if you pick a large enough number $x$ , there are many starting points $d$ such that the $k$ neighborhoods in a row— $d+1, d+2, \dots, d+k$ —are all incredibly messy (have huge class numbers).

The Ingredients of the Proof

To prove this, the authors had to mix three main ingredients:

1. The "Consecutive" Trick (The Garden Analogy)
Imagine you want to plant a row of $k$ flowers, and you want every single one of them to be a rare, giant variety. You can't just pick any random spot; you have to pick a very specific spot in the garden.
The authors use a clever mathematical "sieve" (a filtering tool) to find a specific starting number $d$ . This number is chosen so that when you look at the neighbors $d+1, d+2, \dots, d+k$ , they all share a special property: they are all "split" in a way that makes them prone to being messy. It's like finding a specific patch of soil where, if you plant a seed, the next $k$ seeds you plant will all grow into giants.

2. The "Langlands" Crystal Ball (The Prediction)
To prove these neighborhoods are messy, the authors need to predict their behavior. They rely on a famous, unproven idea called the Langlands Conjecture.
Think of the Langlands Conjecture as a super-accurate crystal ball. It connects two different worlds of math: the world of number fields (our neighborhoods) and the world of "automorphic forms" (complex waves and patterns). By looking at these waves, the authors can predict that the class numbers will be huge. They don't prove the crystal ball works; they assume it does, and then show that if it works, their result follows.

3. The "Regulator" Speed Limit (The Traffic Jam)
There is a catch. The size of the class number depends on two things: how "big" the neighborhood is (its discriminant) and how "fast" the numbers in it can move around (its regulator).

The Discriminant is like the size of the city.
The Regulator is like the speed limit. If the speed limit is too high, the chaos (class number) gets diluted.

For simple cases (like square roots or cube roots), mathematicians already knew the speed limit was low enough to guarantee chaos. But for higher roots (like 5th roots, 7th roots, etc.), nobody knew for sure if the speed limit was low enough.

The Assumption: The authors had to make a guess (Hypothesis 1) that the speed limit for these higher-root neighborhoods is low enough. They ran computer simulations on a supercomputer to check this guess for 5th roots, and it held up.
The Result: If this guess is true, then the "messiness" (class number) grows as fast as the city gets big.

The Bottom Line

The paper proves that consecutive messy neighborhoods exist.

What they did: They showed that for any number of consecutive fields you want (say, 10 in a row), there are infinitely many starting points where all 10 fields have huge class numbers.
The Catch: This proof relies on two things:
1. The Langlands Conjecture (a big, unproven theory in math) being true.
2. A specific guess about the Regulator (the speed limit) being true.

They didn't just say "it's possible"; they gave a formula for how many of these starting points exist. They showed that if you look at all numbers up to a huge number $x$ , you will find at least $x^{1/l}$ of them that start a row of $k$ messy fields. (For example, if you are looking at 5th roots, you find roughly the 5th root of $x$ such starting points).

Why It Matters (In Math Terms)

This is a significant step forward in understanding the "distribution" of complexity in the number system. Before this, we knew messy neighborhoods existed, and we knew we could find them in groups for simple cases (like square roots). This paper extends that knowledge to much more complex types of numbers (higher roots), provided we accept certain mathematical "rules of the road" (the conjectures).

In short: They found a way to guarantee that you can find a whole street of chaotic number neighborhoods, provided the universe follows certain mathematical laws.

Technical Summary: Consecutive Pure Fields of the Form $\mathbb{Q}(\sqrt[l]{a})$ with Large Class Numbers

Problem Statement
The paper addresses the problem of establishing the existence of consecutive pure number fields with arbitrarily large class numbers. While previous works have successfully demonstrated this for quadratic fields ( $l=2$ ) and pure cubic fields ( $l=3$ ), specifically showing the existence of at least $x^{1/2-o(1)}$ and $x^{1/3-o(1)}$ integers $d \le x$ respectively, the generalization to pure fields of prime degree $l \ge 3$ remained open. The authors seek to prove that for a fixed positive integer $k$ and a rational prime $l \ge 3$ , there exist at least $x^{1/l-o(1)}$ integers $1 \le d \le x$ such that the consecutive pure fields $\mathbb{Q}(\sqrt[l]{d+1}), \dots, \mathbb{Q}(\sqrt[l]{d+k})$ possess arbitrarily large class numbers.

Methodology and Assumptions
The proof relies on a combination of analytic number theory, the Langlands program, and specific assumptions regarding the size of regulators.

Langlands Conjecture: The authors invoke the Langlands conjecture (specifically [Bum97, Conjecture 1.8.1]) to associate Artin $L$ -functions with automorphic representations. This allows them to express the class number formula in terms of $L(1, \tilde{\pi}_{d_K})$ , where $\tilde{\pi}$ is an automorphic representation associated with the Galois closure of the pure field.
Hypothesis 1 (Regulator Bound): A critical component of the methodology is Hypothesis 1, which posits an upper bound for the regulator $R_{\mathbb{Q}(\sqrt[l]{a})}$ of fields where $a = n^l + r$ (with $r | l n^{l-1}$ ). The hypothesis states:
$R_{\mathbb{Q}(\sqrt[l]{a})} = o\left(\sqrt{D_{\mathbb{Q}(\sqrt[l]{a})} \log \log D_{\mathbb{Q}(\sqrt[l]{a})}}\right)$
The authors note that while Landau's bound provides a larger estimate ( $O(\sqrt{D} \log^{l-1} D)$ ), it is insufficient to prove arbitrarily large class numbers for $l > 3$ . They provide computational evidence for $l=5$ supporting Hypothesis 1 but acknowledge that a rigorous proof of this bound remains an open problem.
Zero-Free Regions: Using analogues of results by Duke and Daileda, the authors establish zero-free regions for the relevant $L$ -functions, allowing for the approximation of $\log L(1, \tilde{\pi})$ via Dirichlet series truncated at $(\log D)^\epsilon$ .
Sieve Methods and Residue Classes: The proof constructs specific integers $d$ using a sieve argument. By selecting $d$ in specific residue classes modulo a product of primes $q$ , the authors ensure that for the resulting fields, the discriminant $D_j$ satisfies specific splitting conditions (i.e., $D_j$ is an $l$ -th power residue modulo certain primes). This forces the character sums in the $L$ -function expansion to be large.

Key Contributions and Results
The paper presents Theorem 2, which serves as the main result. Under the assumption of the Langlands conjecture and Hypothesis 1, the authors prove:

For a fixed integer $k$ and prime $l \ge 3$ , there exist at least $x^{1/l-o(1)}$ integers $1 \le d \le x$ such that the consecutive pure fields $\mathbb{Q}(\sqrt[l]{d+1}), \dots, \mathbb{Q}(\sqrt[l]{d+k})$ have arbitrarily large class numbers.
Specifically, the class numbers satisfy a lower bound of the form:
$h_{\mathbb{Q}(\sqrt[l]{d+j})} \gg_k \frac{\sqrt{d} \log \log d}{R_{\mathbb{Q}(\sqrt[l]{d+j})}}$
Under Hypothesis 1, this ratio grows arbitrarily large as $d \to \infty$ .

The paper also provides Proposition 2.1, an approximation of $L(1, \tilde{\pi}_{d_K})$ with at most $O(x^{1/4})$ exceptions, and Lemmas 3.1 and 3.2, which establish the existence of suitable residue classes to ensure the necessary splitting behavior of primes in the pure fields.

Significance and Claims
The authors position this work as a generalization of recent results by Byeon and Yhee (2024) on pure cubic fields and earlier work by Cherubini et al. (2023) on quadratic fields. The significance lies in extending the density result ( $x^{1/l-o(1)}$ ) to arbitrary prime degrees $l \ge 3$ .

The paper is modest regarding the unconditional nature of the result. The authors explicitly state that the proof depends on:

The Langlands conjecture for the relevant automorphic representations.
Hypothesis 1, a specific conjecture on the upper bound of regulators for pure fields of the form $\mathbb{Q}(\sqrt[l]{n^l+r})$ .

The authors emphasize that without Hypothesis 1, the known Landau bound for regulators is too weak to guarantee arbitrarily large class numbers for $l > 3$ . They suggest that proving Hypothesis 1 (or a sharper version of it) is a necessary step to make the result unconditional. The paper includes computational evidence for $l=5$ to support the plausibility of Hypothesis 1, noting that the ratio of the regulator to the conjectured bound is significantly less than 1 in the tested cases.

The Big Discovery

The Ingredients of the Proof

The Bottom Line

Why It Matters (In Math Terms)

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