A short remark on the $\ell$-torsion part of class groups

This paper answers a question posed by Ellenberg regarding primitive elements of small height and uses this result to improve Heath-Brown's upper bounds for the $\ell$-torsion part of class groups in purely cubic and general pure number fields of arbitrary odd degree.

The Gist

Imagine you are a detective trying to solve a mystery about a hidden city called Number Field. This city is built on a grid of numbers, and like any city, it has a "population" of special groups called Class Groups.

Within these groups, there are mischievous troublemakers called $\ell$-torsion elements. Think of them as spies who can only cause trouble if they multiply themselves by a specific number ($\ell$) to become invisible (or "zero"). The bigger the city (measured by its "discriminant," $D_K$), the more spies you might expect to find.

For decades, mathematicians had a rough rule of thumb for how many spies could be hiding:

"The number of spies is roughly the square root of the city's size, divided by a tiny fraction."

In 2008, a mathematician named Ellenberg had a brilliant idea. He thought: "Wait a minute! If we can find enough 'primitive' citizens (people who can generate the whole city) who are very small and simple, maybe we can prove there are actually fewer spies than we thought."

He proposed a strategy: Count the small citizens. If there are enough of them, the spy count drops. But he hit a wall. He didn't know if there were enough small citizens to make the math work, especially for certain types of cities. He asked the math community: "Do these small citizens exist in sufficient numbers?"

Enter Martin Widmer, the author of this paper. He steps in to answer that question and refine the detective's tools.

Here is the breakdown of what he found, using some everyday analogies:

1. The "Small Citizen" Problem (Answering Ellenberg)

Ellenberg's strategy relied on finding a specific type of citizen: a primitive element with a small height.

• The Metaphor: Imagine "height" as the complexity of a person's name. A "small height" means a very simple name (like "Bob" instead of "Bartholomew the Third").
• The Question: Are there enough people named "Bob" to prove the spy count is low?
• Widmer's Discovery: For many types of cities (specifically when the spy multiplier $\ell$ is large), the answer is no. There aren't enough "Bobs." The simple citizens are too rare.
• The Result: This means Ellenberg's original, direct strategy doesn't work for those cases. The "spy count" remains higher than he hoped. It's a bit of a bummer, but it's an honest answer to the question.

2. The "Pure" Cities (The Real Win)

However, Widmer didn't just stop at saying "it doesn't work." He looked at a special type of city called a Pure Field.

• The Metaphor: A "Pure Field" is like a city built with a very specific, clean blueprint (e.g., $x^3 - a = 0$). It's not a messy, random city; it has a rigid structure.
• The Old Rule: For these cities, the best previous detective (Heath-Brown) had a rule that said: "The spy count is roughly the square root of the city size, divided by 4."
• Widmer's Upgrade: Widmer looked closer at the "blueprint" of these pure cities. He realized that because the city is so structured, the "citizens" (generators) are actually a bit more complex than we thought, which forces the spies to be even more scarce.
• The New Rule: He improved the math to say: "The spy count is roughly the square root of the city size, divided by 3."
  • Why is this a big deal? In math, dividing by 3 is a much tighter, stronger bound than dividing by 4. It means we are much more confident that there are fewer spies than we used to think.

3. How He Did It (The "Height" Trick)

To get this better result, Widmer used a clever trick involving heights again.

• He proved that in these "Pure Cities," the simplest citizens (the ones with the smallest names) are actually taller (more complex) than the old rules assumed.
• The Analogy: Imagine you thought the shortest person in the city was 5 feet tall. If you find out the shortest person is actually 6 feet tall, you realize the "average" height of the population is higher.
• Because the "shortest" citizens are taller, it becomes harder for the spies to hide among them. This mathematical "tallness" allows Widmer to tighten the net and catch more potential spies, proving there are fewer of them.

Summary

• The Bad News: For general cities, Ellenberg's idea of using "small citizens" to lower the spy count doesn't quite work when the spies are numerous. The small citizens just aren't there in big enough numbers.
• The Good News: For special, structured "Pure Cities" (like cubic fields), Widmer found a way to tighten the rules. He proved that the number of spies is significantly lower than previously thought, improving the best-known mathematical bound.

In a nutshell: Widmer checked the math on a famous detective's theory, found a flaw in the general case, but then used a magnifying glass on a specific type of city to find a much better solution than anyone had before. He didn't just answer a question; he upgraded the detective's toolkit.

Technical Summary

Here is a detailed technical summary of the paper "A Short Remark on the $\ell$-Torsion Part of Class Groups" by Martin Widmer.

1. Problem Statement

The paper addresses the problem of establishing unconditional upper bounds for the size of the $\ell$-torsion subgroup of the ideal class group, denoted as $\#\text{Cl}_K[\ell]$, for number fields $K$ of fixed degree $d$.

• Context: The standard bound, derived from the work of Ellenberg and Venkatesh (2008), relies on the existence of sufficiently many small prime ideals that split completely (or have specific splitting behavior) in $K$. Under the Generalized Riemann Hypothesis (GRH), this yields:
  $$\#\text{Cl}_K[\ell] \ll_{d,\ell,\epsilon} D_K^{1/2 - 1/(2\ell(d-1)) + \epsilon}$$
  where $D_K$ is the absolute discriminant.
• The Gap: Ellenberg (2008) suggested that the proof of the "Key-Lemma" could yield a stronger bound of the form $D_K^{1/2 - f(\ell, d) + \epsilon}$, where $f(\ell, d)$ depends on the distribution of primitive elements of small height. However, it was unclear whether $f(\ell, d)$ could exceed the baseline $1/(2\ell(d-1))$.
• Specific Questions:
  1. What is the precise value of the function $f(\ell, d)$?
  2. Can the bound for pure fields (fields of the form $K = \mathbb{Q}(a^{1/d})$) be improved unconditionally, specifically for cubic fields and general odd degrees?

2. Methodology

Widmer employs a combination of analytic number theory, height theory, and algebraic number theory. The core methodology involves:

• Refining the Key-Lemma: Utilizing a strengthened version of the Ellenberg-Venkatesh Key-Lemma (Lemma 1), which relates the size of the $\ell$-torsion to the number of primitive elements of small height ($N'_K(X)$) and the minimal height of a primitive generator $\eta(K)$.
• Analyzing Primitive Elements: Investigating the quantity $M_{K,\ell} = \inf_X (X^{-1/\ell}(1 + N'_K(X)))$. The paper proves that for $\ell \geq d/2$, the number of small primitive elements is large enough that this quantity does not decay fast enough to improve the exponent beyond the baseline.
• Height Lower Bounds for Pure Fields: For pure fields $K = \mathbb{Q}(a^{1/d})$, the paper utilizes a refined lower bound for the height of generators, $\eta(K)$. This relies on a result by Dubickas (2023) which improves upon Silverman's classical bound when the generator $a$ has a specific factorization structure (square-free components).
• Prime Ideal Construction: The paper constructs specific families of prime ideals in pure fields using Dedekind's factorization theorem and Dirichlet's Prime Number Theorem to ensure the existence of unramified primes of degree 1 with small norm, satisfying the hypotheses of the Key-Lemma unconditionally.

3. Key Contributions and Results

A. Resolution of Ellenberg's Question (Proposition 1)

The paper answers Ellenberg's question regarding the function $f(\ell, d)$.

• Result: For integers $d > 1$ and $\ell \geq d/2$, the function is exactly:
  $$f(\ell, d) = \frac{1}{2\ell(d-1)}$$
• Implication: This proves that for $\ell \geq d/2$, a direct implementation of Ellenberg's strategy (relying solely on the count of small primitive elements) cannot improve the known upper bounds. The "gap" in the exponent cannot be closed by this specific method for these parameters.

B. Improved Bounds for Pure Fields (Proposition 2 & 3)

For pure fields $K = \mathbb{Q}(a^{1/d})$ where $d$ is odd, the paper derives unconditional improvements over the standard bound (1.3) by exploiting the specific arithmetic structure of the generator $a$.

• General Bound (Proposition 3): The paper establishes a bound involving the square-free components $A_i$ of $a$ (where $a = \prod A_i^i$):
  $$\#\text{Cl}_K[\ell] \ll_{\epsilon, \ell, d} D_K^{1/2 + \epsilon} \left( \min_{\frac{d+1}{2} \leq m \leq d-1} \prod_{i=1}^{d-1} A_i^{\frac{i \cdot m}{d} - \lfloor \frac{i \cdot m}{d} \rfloor} \right)^{-1/\ell}$$
• Specific Improvements:
  • Square-free $a$: If $a$ is square-free, the bound improves to:
    $$D_K^{1/2 - \frac{1}{2\ell(d-1)} - \frac{1}{2d\ell(d-1)} + \epsilon}$$
  • Cube-free $a$ (Cubic fields, $d=3$): If $K = \mathbb{Q}(a^{1/3})$ with $a = A_1 A_2^2$ (where $A_1, A_2$ are coprime square-free integers), the bound becomes:
    $$\#\text{Cl}_K[\ell] \ll_{\epsilon, \ell} D_K^{1/2 - 1/(3\ell) + \epsilon} A_2^{1/(3\ell)}$$
    By swapping $A_1$ and $A_2$ (via $a \to a^2/A_2^3$), one can assume $A_2 \leq A_1$, leading to a bound strictly better than Heath-Brown's previous result ($D_K^{1/2 - 1/(4\ell)}$) in many cases, particularly when $A_1$ is large relative to $A_2$.

4. Significance

• Clarification of Limits: The paper definitively shows the limitations of the "small height" strategy for $\ell \geq d/2$, preventing researchers from pursuing a fruitless path for improving bounds in these specific regimes using only the original Ellenberg-Venkatesh framework.
• Unconditional Progress: It provides the first unconditional improvements for the $\ell$-torsion in pure cubic fields (and general pure odd-degree fields) that do not rely on GRH.
• Methodological Advancement: By integrating Dubickas's height lower bounds with the Key-Lemma, the paper demonstrates how the arithmetic structure of the field generator (specifically the factorization of $a$) can be leveraged to tighten class group bounds.
• Connection to Recent Work: The results complement recent breakthroughs (e.g., by Chan and Koymans) that use refined quantities to improve bounds, showing that while the original strategy has limits, adapted versions using specific field structures can still yield significant gains.

In summary, Widmer's paper serves as a critical "reality check" on the theoretical limits of a major strategy in the field, while simultaneously providing concrete, unconditional improvements for a specific and important class of number fields.