🔢 mathematics

Modular curves and bad reduction

This paper establishes that elliptic curves over specific number fields corresponding to points on modular curves, such as those with cyclic torsion of order 20 over $\mathbb{Q}(\sqrt{-11})$ or $\mathbb{Q}(\sqrt{17})$ , necessarily exhibit bad reduction at the prime 3, utilizing distinct proof techniques depending on whether 3 splits or remains inert in the field.

Original authors: Adam Logan, David McKinnon

Published 2026-04-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adam Logan, David McKinnon

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 about a very special type of mathematical object called an elliptic curve. These curves are like intricate, twisting rollercoasters that exist in the world of numbers.

Usually, mathematicians look at two different features of these rollercoasters separately:

The "Torsion" (The Loop): This is a specific, repeating pattern the curve makes. Think of it as a loop-de-loop that the rollercoaster car can get stuck in and spin around a certain number of times before continuing.
The "Bad Reduction" (The Crash): This happens when the rollercoaster hits a rough patch of track at a specific location (a prime number). Instead of gliding smoothly, the track breaks or crumbles.

For a long time, mathematicians thought these two things had nothing to do with each other. They thought you could have a perfect loop-de-loop (torsion) on a track that was perfectly smooth everywhere, or a broken track with no loops.

The Big Discovery
Adam Logan and David McKinnon, the authors of this paper, found a surprising rule: Sometimes, if your rollercoaster has a specific, large loop-de-loop, it is mathematically impossible for the track to be smooth at certain locations. If the loop exists, the crash must happen.

The Magic Map: The Modular Curve

To prove this, the authors use a "Magic Map" called a Modular Curve (specifically named $X_0(20)$ in their main example).

Think of this map as a giant catalog or a menu. Every single elliptic curve in the universe that has a specific type of loop (in this case, a loop of size 20) is listed on this map. If you pick a point on the map, it tells you exactly what that rollercoaster looks like.

The authors discovered that this specific map has a very special shape: it's a genus 1 curve. In simple terms, this means the map itself is shaped like a donut (a torus). This shape is crucial because it allows them to use powerful tools from the "Mordell-Weil group" (a fancy way of saying "the group of points you can add together on the map").

The Detective Work: How the Proof Works

Here is the analogy of their logic:

The "Infinity" Spot: On this Magic Map, there are special points called "cusps." Imagine these as the "end of the line" or the "infinity" spots on the map. If a rollercoaster corresponds to a point near "infinity," it means the track is broken (bad reduction) at a specific prime number (like the number 3).
The Trap: The authors proved that for certain number fields (like $\mathbb{Q}(\sqrt{-11})$ ), every single point on this Magic Map that represents a valid rollercoaster is mathematically forced to be "close" to one of these "infinity" spots when you look at it through the lens of the number 3.
The Conclusion: Because every valid rollercoaster is forced to be near the "infinity" spot, every single one of them must have a broken track (bad reduction) at the number 3.

Why the Number 3 Matters

The paper highlights two different scenarios involving the number 3, which acts like a key to unlock the door:

Scenario A (Splitting): In some number worlds, the number 3 splits into two separate pieces (like a fork in the road). The authors show that if your loop is size 20, you are forced to take a path that leads to a crash at both pieces of the fork.
Scenario B (Inert): In other number worlds, the number 3 stays as one solid block (it doesn't split). The authors use a slightly different trick (involving "twists" of the map) to show that even here, the loop forces a crash.

The "Genus" Problem

The authors mention a catch. Their method works best when the Magic Map is shaped like a donut (Genus 1).

Donut (Genus 1): You can have infinitely many rollercoasters, and the rule applies to all of them.
Higher Shapes (Genus 2+): If the map is shaped like a pretzel or something more complex, the "Finiteness Theorem" kicks in. This means there are only a finite number of rollercoasters on the map. So, while the rule still works, it only applies to a small, limited list of curves, not an infinite family.

Real-World Takeaway

The paper concludes with a list of specific "Levels" (sizes of loops) and "Primes" (locations of crashes).

The Main Result in Plain English:
"If you find an elliptic curve (a rollercoaster) defined over a specific number field that has a repeating loop of size 20, you can be 100% certain that the track will break at the number 3. It doesn't matter how you try to build it; the math forces it to crash there."

They also provide a recipe for other loop sizes (like 24, 32, etc.) and other crash locations (like 5, 7, 11), showing that this is a widespread phenomenon, not just a one-time fluke.

In summary: The paper connects two seemingly unrelated features of mathematical curves (their loops and their broken tracks) by using a special "donut-shaped" map. They prove that having a big loop forces the track to break at specific numbers, turning a mystery into a predictable rule.

1. Problem Statement

The paper addresses a fundamental question in the arithmetic of elliptic curves: the relationship between the torsion structure of an elliptic curve and its places of bad reduction.

Context: Traditionally, results in this area proceed from bad reduction to torsion (e.g., if a curve has additive reduction at a prime $p$ , its rational torsion is a $p$ -group).
The Gap: The authors investigate the converse direction: Can specific torsion structures force the existence of bad reduction at specific primes?
Specific Focus: The paper focuses on elliptic curves $E$ defined over a number field $k$ that possess a rational cyclic torsion subgroup of a specific order (e.g., order 20). The goal is to prove that such curves must have bad reduction (specifically, not even potentially good reduction) at certain primes $\mathfrak{p}$ of $k$ .

2. Methodology

The core methodology relies on the geometry of modular curves, specifically $X_0(N)$ and $X_1(N)$ , which parameterize elliptic curves with specific torsion structures.

Modular Curve Correspondence: An elliptic curve $E/k$ with a cyclic torsion subgroup of order $N$ corresponds to a $k$ -rational point $P$ on the modular curve $C = X_0(N)$ (or $X_1(N)$ ).
Genus 1 Constraint: The authors restrict their primary analysis to modular curves of genus 1. This is crucial because:
- The set of rational points on a genus 1 curve forms a finitely generated abelian group (Mordell-Weil group).
- If the curve has positive rank, there are infinitely many isomorphism classes of elliptic curves to consider.
Reduction Modulo $\mathfrak{p}$ : The proof strategy involves analyzing the reduction of the modular curve $C$ $C$ modulo a prime $\mathfrak{p}$ $p$ .
- Cusps as Torsion: The cusps of the modular curve correspond to degenerate elliptic curves (with $j$ -invariant $\infty$ ). These points are torsion points in the Mordell-Weil group of $C$ .
- The "Locus at Infinity": If a point $P \in C(k)$ reduces modulo $\mathfrak{p}$ to a cusp, the corresponding elliptic curve $E_P$ has a non-integral $j$ -invariant at $\mathfrak{p}$ , implying bad reduction (specifically, multiplicative or additive reduction that is not potentially good).
Surjectivity Arguments: The authors use group-theoretic arguments to show that every $k$ $k$ -rational point on the modular curve must reduce to a cusp modulo $\mathfrak{p}$ $p$ . This is achieved by:
1. Identifying generators of the Mordell-Weil group $C(k)$ .
2. Showing that these generators (and the torsion subgroup) reduce to cusps.
3. Proving that the set of points reducing to cusps forms a subgroup, thereby covering the entire group $C(k)$ .
2-Descent: For specific cases involving quadratic fields and the curve $X_0(20)$ , the authors employ 2-descent techniques to analyze the surjectivity of the map $E(\mathbb{Q}) \oplus E^\sigma(\mathbb{Q}) \to E(K)$ , where $K$ is a quadratic extension and $E^\sigma$ is the quadratic twist.

3. Key Contributions and Results

A. General Theorems on Bad Reduction

The paper establishes several theorems linking torsion to bad reduction:

Theorem 2.1 & 2.3: If the modular curve $C$ has genus 1 and every point in the Mordell-Weil group $C(k)$ reduces to a cusp modulo a prime $\mathfrak{p}$ , then every elliptic curve over $k$ with the corresponding torsion structure has bad reduction at $\mathfrak{p}$ .
Theorem 2.4: A refined result for quadratic fields $K = \mathbb{Q}(\sqrt{d})$ where a prime $p$ is inert. If the map from the rational points of the curve and its twist to the points over $K$ is surjective, and reduction conditions are met, bad reduction is forced.

B. Specific Case: Order 20 Torsion

The paper provides concrete examples using the modular curve $X_0(20)$ , which has genus 1 and 6 cusps.

Theorem 2.2: If an elliptic curve over a number field $k$ $k$ has a rational cyclic torsion subgroup of order 20, and the prime 3 is totally split in $k$ $k$ , then the curve has bad reduction at all primes lying over 3.
- Example: Over $k = \mathbb{Q}(\sqrt{-11})$ , 3 splits. Thus, any such curve has bad reduction at primes over 3.
Theorem 2.2 (Inert Case): If 3 is inert in $k$ (e.g., $k = \mathbb{Q}(\sqrt{17})$ ), the result still holds under specific conditions regarding the generators of the Mordell-Weil group.

C. Classification of Levels and Primes

The authors systematically analyze modular curves $X_0(N)$ of genus 1 for various levels $N$ .

Levels with Non-Cuspidal Points: For $N \in \{11, 14, 15, 17, 19, 21, 27\}$ , there exist non-cuspidal rational points. The authors note their techniques generally fail here because not all points reduce to cusps.
Levels with Success: For $N \in \{20, 24, 32, 36, 49\}$ $N \in {20, 24, 32, 36, 49}$ , the conditions are satisfied.
- Table 1 lists specific primes $p$ for each $N$ where the hypothesis holds.
- Example: For $N=24$ , any elliptic curve over $\mathbb{Q}(\sqrt{10})$ with a subgroup of order 24 must have bad reduction at 5 (since 5 is ramified in $\mathbb{Q}(\sqrt{10})$ ).

D. Application to $X_1(N)$

The paper extends the analysis to $X_1(N)$ curves (parameterizing points of order $N$ ).

They analyze $X_1(11), X_1(14), X_1(15)$ and determine specific primes (e.g., 2, 3, 5) where bad reduction is forced for curves with points of order 11, 14, or 15.
They note that results are not invariant under isogeny due to changes in cusp structures.

E. Divisibility and Rank Analysis (Section 3)

The authors provide a rigorous proof (Theorem 3.1) regarding the surjectivity of the map $E(\mathbb{Q}) \oplus E^\sigma(\mathbb{Q}) \to E(\mathbb{Q}(\sqrt{q}))$ for the curve $X_0(20)$ and quadratic fields.

Using 2-descent, they determine the structure of the Selmer group.
Corollary 3.2: They derive a condition on a prime $q$ such that every elliptic curve over $\mathbb{Q}(\sqrt{q})$ with a cyclic subgroup of order 20 has bad reduction at 3. This unifies the split and inert cases of 3 in the base field.

4. Significance

Reversing the Logic: The paper successfully reverses the standard direction of inference in the study of elliptic curves, moving from torsion constraints to reduction properties.
Infinite Families: Unlike results relying on Faltings' Theorem (which only apply to finitely many curves in higher genus), the genus 1 focus allows the authors to make statements about infinitely many isomorphism classes of elliptic curves over specific number fields.
Explicit Criteria: The work provides explicit, checkable criteria (involving Legendre symbols and splitting behavior of primes) to determine if a curve with specific torsion must have bad reduction.
Methodological Bridge: It bridges the gap between the arithmetic of modular curves (Mordell-Weil groups, cusps) and the local arithmetic of elliptic curves (reduction types, $j$ -invariants).

5. Conclusion

Logan and McKinnon demonstrate that the geometry of modular curves of genus 1 imposes strict constraints on the reduction behavior of the elliptic curves they parameterize. By proving that all rational points on these modular curves reduce to cusps modulo specific primes, they establish that elliptic curves with certain torsion structures cannot have potentially good reduction at those primes. This yields powerful, general results for infinite families of curves, particularly for torsion orders like 20, 24, and 32.