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Isolated points on modular curves

This paper establishes a single-sink theorem and a uniform strategy for classifying isolated points on modular curves $X_H$ , applying these results to determine isolated points with rational $j$ -invariants on level 7 curves and $X_0(n)$ while developing a broader theory of isolated divisors on geometrically disconnected varieties.

Original authors: Kenji Terao

Published 2026-03-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Kenji Terao

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 massive mystery involving millions of suspects. In the world of mathematics, specifically number theory, these "suspects" are special numbers called $j$ -invariants. These numbers describe the shapes of objects called elliptic curves (which look like twisted donuts).

The paper you are asking about is a guidebook written by Kenji Terao on how to find the "Isolated Points" among these suspects.

The Big Picture: The "Isolated Point" Mystery

Think of a Modular Curve as a giant, complex map or a city.

The City (Modular Curve): This city is filled with points. Most points in this city are part of huge, predictable neighborhoods. If you find one person there, you can easily find thousands more just by following a simple pattern (like a family tree or a repeating street grid). Mathematicians call these "Parametrized Points." They are easy to find because they come in infinite families.
The Isolated Point: Now, imagine a single person standing alone in the middle of a field, far away from any neighborhood. There is no pattern to find them, no family tree to follow, and no infinite crowd of friends nearby. They are truly isolated.

The mystery of the paper is: How do we find these lonely, isolated people in these massive mathematical cities? And more importantly, how do we know if they exist at all?

The Detective's Toolkit: The "Single-Sink" Theorem

Before this paper, finding these isolated points was like searching for a needle in a haystack, where the haystack was constantly growing. You had to check every single city (every modular curve) one by one.

Terao introduces a brilliant new tool called the Single-Sink Theorem. Here is the analogy:

Imagine you are looking for a specific type of lost dog (an isolated point with a specific $j$ -invariant).

The Old Way: You check every park, every street, and every house in the entire country.
The New Way (Single-Sink): Terao proves that if this lost dog exists anywhere in the country, it must be hiding in one specific, very special kennel (a specific curve called $X_G$ ).

This is the "Single Sink." No matter how many different cities (modular curves) you look at, if an isolated point exists, all the clues point back to this one single location.

Why is this amazing? It turns an impossible task (searching infinite cities) into a manageable one (checking just one specific kennel). If the dog isn't in that one kennel, it doesn't exist anywhere.

The Strategy: The "Funnel" Method

The paper outlines a three-step strategy to solve these mysteries, which the author calls a "uniform strategy."

The Funnel (Narrowing the Search):
Instead of checking every possible number, the author uses the "Single-Sink" idea to filter out 99.9% of the possibilities. It's like using a sieve to separate the gold from the sand. We only keep the numbers that could possibly be isolated.
The Map (Reducing the Cities):
Once we have a small list of suspect numbers, we don't need to check every city again. The paper proves that if an isolated point exists in a huge city, it must also exist in a tiny, manageable village (a curve of a specific, small level). This shrinks the search area from a continent down to a single neighborhood.
The Final Check:
With the search area now tiny, the author uses computer programs (like a high-tech magnifying glass) to check the remaining spots.

Real-World Examples in the Paper

The author puts this detective work to the test with two specific cases:

Case 1: Level 7 Curves.
Think of this as investigating a specific district of the city. The author found that there is exactly one special number ( $j = \frac{3^3 \cdot 5 \cdot 7^5}{2^7}$ ) that creates isolated points in this district. They identified exactly where these points hide.
Case 2: The $X_0(n)$ Curves.
This is a much larger, more chaotic part of the city. The author couldn't prove everything with 100% certainty (because some mathematical rules are still just guesses), but they made a very strong prediction. They listed a specific set of numbers (like $-121$, $-24729001$, etc.) that are the only candidates for being isolated points. It's like saying, "If there are any lonely people in this district, they must have one of these 15 specific names."

The "Geometrically Disconnected" Twist

There is one tricky part of the paper. Some of these mathematical cities are not one big connected landmass; they are like archipelagos (groups of islands).

In math terms, these are geometrically disconnected varieties.
The author had to invent a new way to count "people" on these islands. They developed a theory of "Isolated Divisors" (a fancy way of saying "groups of points that act like a single isolated unit").
The Metaphor: Imagine trying to find a lost hiker on a group of islands. If you only look at one island, you might miss them. Terao's theory allows you to look at the whole archipelago as a single unit, ensuring you don't miss the hiker just because they are on a different island.

Why Does This Matter?

You might ask, "Who cares about lonely points on abstract curves?"

Efficiency: It saves mathematicians from doing millions of useless calculations. Instead of checking infinite possibilities, they check one.
Understanding the Universe: These curves are deeply connected to the fundamental structure of numbers. Finding the "isolated" ones helps us understand the exceptions to the rules of the universe.
The "Mazur's Program B": This connects to a famous problem about what shapes elliptic curves can take. Terao's method provides a direct pipeline to solve parts of this huge, decades-old puzzle.

Summary

In simple terms, Kenji Terao wrote a manual on how to stop searching the whole ocean for a specific fish. He proved that if the fish exists, it's hiding in a specific, small pond. He then gave a step-by-step guide to drain that pond and catch the fish, successfully finding them in a few specific cases and predicting where to look for the rest.

It's a story of finding order in chaos by realizing that all the chaos points back to a single, manageable source.

1. Problem Statement

The paper addresses the classification of isolated points on modular curves $X_H$ , where $H$ is a subgroup of $GL_2(\mathbb{Z}/n\mathbb{Z})$ (or an open subgroup of $GL_2(\hat{\mathbb{Z}})$ ).

Context: A point on a curve defined over a number field is "isolated" if it does not belong to an infinite family of points of the same degree. Such families arise either from maps to $\mathbb{P}^1$ (P1-parametrized) or from positive-rank abelian subvarieties of the Jacobian (AV-parametrized).
Gap: Previous work (e.g., by Bourdon, Ejder, Liu, Odumodu, and Viray) focused almost exclusively on the specific family of modular curves $X_1(n)$ . There was no unified framework for studying isolated points on general modular curves $X_H$ (including $X_0(n)$ , $X_s(p)$ , etc.), particularly when these curves are geometrically disconnected.
Goal: To develop a general theory for isolated points on arbitrary modular curves $X_H$ and apply it to classify non-CM isolated points with rational $j$ -invariants on curves of level 7 and on the family $X_0(n)$ .

2. Methodology

The author employs a multi-layered approach combining algebraic geometry, group theory, and Galois representations.

A. Theoretical Foundation: Isolated Divisors

To handle geometrically disconnected curves (a common feature of $X_H$ ), the author generalizes the concept of isolated points to isolated divisors on geometrically disconnected varieties.

Stein Factorization: The paper utilizes the Stein factorization $X \to \text{Spec}(K) \to \text{Spec}(k)$ to relate the geometry of a disconnected curve $X$ over $k$ to a connected curve over an extension field $K$ .
Divisor Schemes: The theory of divisor schemes ( $\text{Div}_{X/k}$ ) and Picard schemes ( $\text{Pic}_{X/k}$ ) is extended to the disconnected case using Weil restrictions.
Mapping Theorems: Two key theorems are established regarding morphisms $f: C \to D$ $f : C \to D$ :
1. Forward Map: If $x \in C$ is isolated and $\deg(x) = \deg(y) \cdot \deg(f)$ , then $y = f(x)$ is isolated.
2. Backward Map: If $y \in D$ is isolated and $\deg(x) = \deg(y)$ , then $x \in f^{-1}(y)$ is isolated.
  These theorems allow the propagation of "isolation" properties between different modular curves.

B. The Single-Sink Theorem

The core theoretical contribution is the Single-Sink Theorem.

Statement: For any non-CM isolated point $x$ on a modular curve $X_H$ with $j$ -invariant $j(x)$ , there exists an isolated point $y$ on a specific "sink" curve $X_G$ (where $G$ is the image of the adelic Galois representation of an elliptic curve with $j$ -invariant $j(x)$ ) such that $j(y) = j(x)$ .
Implication: The existence of isolated points with a given $j$ -invariant on any modular curve can be traced back to a single curve determined solely by the Galois image of the associated elliptic curve. This reduces the search space significantly.

C. Computational Strategy

The author proposes a three-step algorithm to find isolated points for a family of curves $\mathcal{S}$ and a set of $j$ -invariants $\mathcal{J}$ :

Restrict $j$ -invariants: Use generalizations of the Single-Sink Theorem to find a finite subset $\mathcal{J}' \subset \mathcal{J}$ containing all possible $j$ -invariants of isolated points. This involves analyzing the possible images of Galois representations.
Restrict Curves: For each $j \in \mathcal{J}'$ , use "dual" theorems to find a finite subset of curves $\mathcal{S}_j \subset \mathcal{S}$ where isolated points could exist. This often reduces the problem to curves of bounded level.
Verification: Explicitly check the remaining pairs $(j, X)$ for isolation using Riemann-Roch dimensions (for P1-isolation) and Mordell-Weil ranks of Jacobians (for AV-isolation).

3. Key Contributions

Theory of Isolated Divisors: A rigorous framework for defining and studying isolated points on geometrically disconnected varieties, resolving technical obstacles that prevented the direct application of previous results to general modular curves.
The Single-Sink Theorem: A powerful structural result linking isolated points across the entire landscape of modular curves to a single "sink" curve defined by the Galois image.
Isolation Graphs: The introduction of a directed graph (the "isolation graph") where vertices represent closed points and edges represent the mapping theorems. This allows for the systematic reduction of the search space by identifying terminal vertices (sinks) and eliminating non-isolated candidates.
Uniform Strategy: A generalizable algorithm for classifying isolated points on any family of modular curves, moving beyond the specific case of $X_1(n)$ .

4. Main Results

Application 1: Modular Curves of Level 7

The paper classifies all non-CM isolated points with rational $j$ -invariant on modular curves of level 7.

Result: There is exactly one such $j$ -invariant:
$j = \frac{3^3 \cdot 5 \cdot 7^5}{2^7}$
Classification: This $j$ -invariant corresponds to isolated points on 9 specific conjugacy classes of subgroups $H \leq GL_2(\mathbb{Z}/7\mathbb{Z})$ .
Method: The authors constructed an isolation graph with 12,690 vertices (reduced to 252 via conjugation symmetries). They proved that only points on the curve $X(7)$ (level 1, genus 3) with degree 18 could be isolated, and verified their isolation explicitly.

Application 2: Modular Curves $X_0(n)$

Assuming Zywina's Conjecture (Conjecture 7.4) regarding the images of adelic Galois representations of elliptic curves over $\mathbb{Q}$ , the paper classifies the $j$ -invariants of non-CM isolated points on $X_0(n)$ .

Result: The set of possible $j$ -invariants is finite and explicitly listed (14 values).
List of $j$ -invariants: Includes values such as $-121$, $-24729001$, $-25/2$ , $3375/2$ , $-9317$, and $-162677523113838677$.
Significance: Unlike the level 7 case, the paper does not specify the exact curve $X_0(n)$ for every $j$ -invariant but proves that if an isolated point exists, its $j$ -invariant must be in this set. The minimal $n$ for each $j$ is provided (e.g., $n=11$ for $j=-121$ , $n=37$ for $j=-9317$ ).

5. Significance

Unification: The paper successfully unifies the study of isolated points across diverse families of modular curves ( $X_1(n)$ , $X_0(n)$ , etc.), providing a single theoretical engine rather than ad-hoc methods for each family.
Handling Disconnectedness: By developing the theory of isolated divisors on geometrically disconnected varieties, the paper removes a major technical barrier in the arithmetic geometry of modular curves.
Connection to Mazur's Program B: The work explicitly links the problem of finding isolated points to Mazur's Program B (classifying images of Galois representations). It demonstrates that progress on Galois image classification directly yields progress on the classification of isolated points.
Computational Feasibility: The "Isolation Graph" method provides a practical, algorithmic approach to reducing infinite search spaces to finite, computable problems, as demonstrated by the successful classification for level 7 and $X_0(n)$ .

In summary, Terao's work provides the necessary theoretical infrastructure and a practical algorithm to systematically identify "exceptional" points on modular curves, extending the scope of previous research from $X_1(n)$ to the broader universe of modular curves $X_H$ .