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Uniform Bounds on S-Integral Torsion Points for $\mathbb{G}_m$ and Elliptic Curves

This paper establishes uniform bounds on the number and degree of $S$ -integral torsion points relative to a non-torsion point for the multiplicative group $\mathbb{G}_m$ and elliptic curves over number fields of bounded degree.

Original authors: Jit Wu Yap

Published 2026-01-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jit Wu Yap

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 detective trying to solve a mystery in the vast, infinite city of Number Theory. In this city, there are two main types of "residents":

The Torsion Points: These are the "perfectly periodic" residents. Think of them like the hands of a clock or the roots of unity (numbers that, when multiplied by themselves enough times, return to 1). They are predictable, finite in their cycles, and they live in specific, neat neighborhoods.
The Non-Torsion Points: These are the "wanderers." They never repeat their pattern. They are like a person walking through the city who never returns to the exact same spot twice.

The Mystery:
The paper investigates a specific rule about how these two types of residents interact. The rule is called S-Integrality.

Think of the city as having a few "special districts" (called S). These districts are the only places where the "perfect" residents (Torsion Points) are allowed to get very close to the "wanderers" (Non-Torsion Points). Everywhere else in the city, there is a strict "social distancing" rule: the periodic residents must stay far away from the wanderers.

The big question the author, Jit Wu Yap, asks is: How many of these periodic residents can break the rules and get close to a wanderer?

The Main Discovery: Uniform Bounds

In the past, mathematicians knew that for any single wanderer, there are only a finite number of periodic residents who can get close to them. But they didn't know if there was a universal "cap" on how many could get close if you looked at all wanderers at once, especially if those wanderers were getting more and more complex (living in larger and larger number fields).

Yap's paper proves that yes, there is a universal cap.

Here is the breakdown using simple analogies:

1. The "Gm" Case (The Circle of Numbers)

Imagine the periodic residents are points on a perfect circle (like the roots of unity).

The Finding: If you have a wanderer who isn't too complicated (their "degree" is bounded), there is a strict limit on how many points on the circle can get close to them, regardless of where the wanderer is.
The Catch: If the wanderer gets extremely complex (living in a very large number field), the number of allowed "close calls" might grow, but Yap proves it grows in a predictable, controlled way. It doesn't explode into infinity.
The "Exception" Rule: Yap finds that for every "special district" (place in the number system), there might be one specific periodic resident that breaks the rules. But once you account for these few exceptions, the rest of the periodic residents are strictly kept at a distance. It's like saying, "In every neighborhood, maybe one person can get close, but no one else can."

2. The Elliptic Curve Case (The Wobbly Donuts)

Now, imagine the residents live on a more complex shape called an Elliptic Curve (imagine a donut shape that can twist and turn).

The Finding: The same logic applies. Even on these complex shapes, the periodic residents (torsion points) cannot crowd the wanderers too closely, except for a few specific exceptions.
The Complex Multiplication Twist: For a special type of donut (one with "Complex Multiplication"), Yap proves a very strict rule: there is a hard limit on how close they can get, with no exceptions needed.
The General Case: For all other donuts, the rule holds, but we have to allow for a few more "exceptions" (a small number of periodic residents that can get close). This is because the math used to prove the distance is slightly less precise for these general shapes.

How Did They Solve It? (The Detective's Toolkit)

Yap didn't just look at one wanderer; he looked at the whole crowd. To do this, he used a powerful mathematical tool called Equidistribution.

The Analogy: Imagine you have a bag of marbles (the periodic residents). If you shake the bag, the marbles spread out evenly across the table.
The Problem: Usually, mathematicians know the marbles spread out evenly eventually. But Yap needed to know how fast they spread out and how the speed changes if the table (the number field) gets bigger.
The Solution: He used advanced techniques (involving "Berkovich spaces," which are like a super-detailed map of the number system) to measure exactly how fast the marbles spread out. He proved that even as the "table" gets huge, the marbles still spread out fast enough to ensure they don't cluster too tightly around the wanderer.

The "Linear Forms" Secret Weapon

To prove the periodic residents can't get too close, Yap used a technique called Linear Forms in Logarithms.

The Analogy: Think of this as a super-precise ruler. If a periodic resident tries to get close to a wanderer, this ruler measures the distance. The ruler says, "If you are this close, you must be a very specific type of resident, and there are only a few of you."
The Limitation: For the general elliptic curves, the ruler wasn't quite sharp enough to catch every rule-breaker, so Yap had to allow for a few "exceptions" (like allowing one person per neighborhood to get close). But for the special "Complex Multiplication" curves, the ruler was sharp enough to catch everyone.

Summary of the Results

Uniformity: The number of "rule-breaking" periodic points is bounded. It doesn't matter how many wanderers you have or how complex they are; there is a limit.
Exceptions: For the most general cases, you have to allow for a small number of exceptions (one per "district" or place in the number system). If you ignore these few exceptions, the rule holds perfectly.
Growth: The limit grows with the complexity of the wanderer, but Yap shows exactly how it grows (polynomially or exponentially depending on the case), proving it stays finite.

In short: The paper proves that in the chaotic world of numbers, the "perfectly repeating" numbers cannot crowd the "non-repeating" numbers too closely. There is a universal law of social distancing, with only a very small, predictable number of exceptions allowed.

Technical Summary: Uniform Bounds on S-Integral Torsion Points for $G_m$ and Elliptic Curves

Problem Statement
The paper addresses the problem of establishing uniform bounds on the number and degree of $S$ -integral torsion points relative to a non-torsion point $\beta$ . Let $K$ be a number field and $S$ a finite set of places including all archimedean places. Two points $\alpha, \beta$ are defined as $S$ -integral relative to each other if their Zariski closures in $\mathbb{P}^1_{\mathcal{O}_K}$ do not intersect outside $S$ .

The foundational result by Baker, Ih, and Rumely [BIR08] (Theorem 1.1) establishes that for a fixed $\beta \in K^\times \setminus \mu_\infty$ , the set of roots of unity $\zeta$ that are $S$ -integral relative to $\beta$ is finite. The paper seeks to strengthen this by proving uniformity: specifically, whether the bounds on the degree or order of such torsion points can be made independent of the specific choice of $\beta$ , provided $\beta$ varies over number fields of bounded degree.

Methodology
The core methodology relies on quantitative equidistribution of Galois orbits of torsion points. The proof strategy involves:

Equidistribution Frameworks:
- For the multiplicative group $G_m$ , the paper utilizes the Berkovich projective line $\mathbb{P}^1_{\text{Berk}, v}$ and the quantitative equidistribution theorems of Favre–Rivera-Letelier [FR06]. This allows for explicit control over the rate of convergence of Galois orbits to the equilibrium measure (Haar measure on the unit circle for archimedean places, and the delta mass at the Gauss point for non-archimedean places) as the base field varies.
- For elliptic curves $E$ , the paper adapts results from Baker–Petsche [BP05] and Petsche [Pet09]. It constructs analogous quantitative estimates for the convergence of torsion orbits to the canonical measure on the Berkovich analytification $E^{\text{an}}_v$ (Haar measure for archimedean, skeleton measure for non-archimedean).
Linear Forms in Logarithms:
To handle the "uniformity" aspect (bounding the distance between a torsion point and a varying $\beta$ ), the paper employs lower bounds for linear forms in logarithms.
- For $G_m$ , it uses the bounds of Laurent, Mignotte, and Nesterenko [LMN95].
- For elliptic curves, it utilizes results from David–Hirata-Kohno [DH09] for the general case and Ably–Gaudron [AG03] for the complex multiplication (CM) case.
- A critical technical step involves bounding the term $\lambda_v(x - \beta)$ (the local height) in terms of the size of the Galois orbit $|F|$ and the height $h(\beta)$ . The paper notes that general elliptic curve bounds contain an extra factor of $\log^+ h(\beta)$ , which obstructs a direct uniform bound, necessitating the allowance of a finite number of exceptional orbits.
Discreteness Arguments:
For non-archimedean places, the paper exploits the discreteness of torsion points in residue classes. It shows that for a fixed $\beta$ , only a bounded number of torsion points can be "too close" to $\beta$ in the $v$ -adic metric, allowing the authors to exclude a small set of exceptions to achieve polynomial bounds.

Key Contributions and Results

1. For the Multiplicative Group ( $G_m$ ):

Theorem 1.2: Establishes a uniform bound on the degree of a root of unity $\zeta$ that is $S$ -integral relative to a non-torsion point $\beta$ , where $\beta$ varies over number fields of degree $\le D$ . Specifically, there exists a constant $C = C([K:\mathbb{Q}], S)$ such that $\deg_{\mathbb{Q}}(\zeta) < C$ .
Theorem 1.3: Provides a stronger result allowing for a finite number of exceptions. For any $\beta$ , the set of $S$ -integral roots of unity with degree exceeding a polynomial bound in $|S|$ and $[K:\mathbb{Q}]$ consists of at most $|S_{\text{fin}}|$ Galois orbits. The bound grows polynomially with the degree of the field, though the general proof yields an exponential bound; the polynomial bound is achieved by allowing exceptions.

2. For Elliptic Curves ( $E$ ):

Theorem 1.4 (CM Case): For an elliptic curve with complex multiplication (CM), the paper proves a uniform bound on the degree of a torsion point $z$ that is $S$ -integral relative to a non-torsion $\beta$ . The CM assumption is crucial here as it allows the removal of the $\log^+ h(\beta)$ factor in the linear forms in elliptic logarithms, enabling a result without exceptions.
Theorem 1.5 (General Case): For a semistable elliptic curve $E$ (without the CM assumption), the paper proves a uniform bound that holds for all but a finite number of Galois orbits. Specifically, the set of $S$ -integral torsion points with degree exceeding a bound involving $|S|^3 [L:K]^{11}$ is a union of at most $|S|$ Galois orbits. The exceptions arise because general bounds on linear forms in elliptic logarithms include a $\log^+ h(\beta)$ factor, and archimedean estimates are not strong enough to rule out all exceptions without this allowance.

Significance and Claims
The paper claims to extend the finiteness results of Baker, Ih, and Rumely [BIR08] into the realm of uniformity. While [BIR08] proved finiteness for a fixed $\beta$ , this work demonstrates that the bounds on the torsion points depend only on the degree of the field containing $\beta$ and the set $S$ , rather than the specific arithmetic properties of $\beta$ itself.

The authors note that their proof for $G_m$ (Theorem 1.2) provides an upper bound growing exponentially with the degree $[K:\mathbb{Q}]$ , whereas a polynomial growth is expected based on constructions by Boyd. They clarify that a polynomial bound is achievable (Theorem 1.3) only if one allows for a small, explicitly bounded set of exceptional Galois orbits.

For elliptic curves, the paper highlights the technical necessity of the Complex Multiplication assumption for Theorem 1.4 to avoid exceptions, attributing this to the current limitations of linear forms in elliptic logarithms for non-CM curves. The results for general elliptic curves (Theorem 1.5) are presented as the best possible uniform bounds under current knowledge, requiring the exclusion of a finite number of orbits to overcome the logarithmic height factors in the underlying Diophantine approximation theorems.

The paper concludes by situating these results within the broader context of Ih's Conjecture regarding the finiteness of $S$ -integral preperiodic points for rational maps, suggesting that the methods developed here, combined with equidistribution theorems, could be extended to Drinfeld modules and potentially to the general conjecture.