On the torsion growth in quadratic number fields for elliptic curves defined over the rationals

This paper investigates the inverse problem of determining the properties of a quadratic number field based on the growth of an elliptic curve's torsion subgroup defined over the rationals, establishing an explicit relationship between the primes dividing the curve's conductor and the conductor of the field extension.

The Gist

Imagine you have a magical, infinite grid of points called an Elliptic Curve. This grid lives on a simple, familiar world called the Rational Numbers (fractions like 1/2, 3/4, -5, etc.).

On this grid, there are special "anchor points" called Torsion Points. These are points that, if you keep adding them to themselves, eventually bring you back to the starting line (the "point at infinity"). The collection of all these anchor points forms a Torsion Subgroup. Think of this subgroup as the "fingerprint" or the "DNA" of the curve's structure over the rational numbers.

The Problem: Expanding the Universe

Now, imagine we decide to expand our universe. We take our curve and move it into a slightly larger, more complex world called a Quadratic Number Field. This is like adding a new dimension, specifically involving the square root of a number (like $\sqrt{2}$ or $\sqrt{-5}$).

When we move the curve to this new world, something interesting happens: New anchor points appear. The fingerprint gets bigger. The torsion subgroup "grows."

For a long time, mathematicians knew the rules for how the fingerprint could grow. They had a table saying, "If you start with Group A, you can end up with Group B, C, or D."

But this paper asks the reverse question:

"We know the fingerprint grew. We know the curve started on the Rational Numbers. Can we look at the growth and tell you exactly what the new world (the Quadratic Field) looks like?"

Specifically, if the fingerprint grows, what can we say about the number $d$ inside the square root $\sqrt{d}$ that defines this new world?

The Detective Work: Following the Clues

The authors act like detectives. They know that for a new anchor point to appear in the new world, the "coordinates" of that point must live there. They use a powerful tool called Galois Representations (think of this as a security camera system that watches how the points move and interact).

They discovered that the "security cameras" only allow certain types of growth. If a new point appears, the number $d$ (the key to the new world) must have specific properties related to the curve's own history.

Here is the breakdown of their findings using simple analogies:

1. The "Bad Neighborhood" Rule (Primes 2, 5, 7)

Every elliptic curve has a "conductor" ($N_E$). Think of the conductor as a list of "bad neighborhoods" where the curve behaves erratically or breaks down.

• The Discovery: If the fingerprint grows because of the number 2, 5, or 7, then the new world ($\sqrt{d}$) must be built using one of those same "bad numbers."
• The Analogy: Imagine you are trying to build a new house (the new world) and you find a strange, new piece of furniture (a new torsion point) inside it. The authors prove that if that furniture is made of "2-wood," "5-wood," or "7-wood," then the land you built the house on ($d$) must also be made of that same type of wood. You can't find a "5-wood" chair in a house built entirely on "3-wood" land.
• The Result: If the curve has good behavior at 2, 5, or 7, but the torsion grows in a field where 2, 5, or 7 is a "bad" number (ramifies), then the curve must have had bad behavior at that number all along.

2. The "Special Exception" (Prime 3)

The number 3 is the rebel of the group.

• The Discovery: Unlike 2, 5, and 7, the number 3 can sometimes cause growth even if the curve behaves perfectly well at 3.
• The Analogy: Imagine 3 is a magical key that can open a door even if the lock (the curve's reduction) is in perfect condition. However, the authors found that if 3 does open the door and the curve is in perfect condition, the growth is very predictable and simple (it just adds a specific, small group).
• The Catch: If the growth is complex (like jumping from no points to a huge group of 9 points), then the curve must have been in a "bad neighborhood" at 3 after all.

The Big Picture: The "Sieve"

The main achievement of this paper is creating a Sieve.

Before this paper, if you saw a curve's torsion grow, you might have to check thousands of possible quadratic fields to see which one caused it.
Now, thanks to this paper, you can run the "Sieve":

1. Look at the curve's "Bad Neighborhood" list (its conductor).
2. Look at the new world's "Bad Neighborhood" list (the prime factors of $d$).
3. The Rule: If the torsion grew, the new world's bad numbers must be either:
   • Numbers that were already bad for the curve (in the conductor), OR
   • The number 3 (with some specific conditions).

Why Does This Matter?

This is like having a map that tells you exactly where you can't go.

• If you are a mathematician trying to find all elliptic curves with a specific property, you don't need to check every single number field. You can instantly discard any field that doesn't fit the "Sieve" rule.
• It connects the geometry of the curve (how it behaves at different numbers) with the algebra of the field (what numbers make up the square root).

Summary in One Sentence

If an elliptic curve's "fingerprint" grows when you move it to a new world made of square roots, the "ingredients" of that new world must be either ingredients the curve already had trouble with, or the special number 3. You can't get a growth in a new world made of "7-ingredients" unless the curve was already struggling with "7" back home.

Technical Summary

Here is a detailed technical summary of the paper "On the Torsion Growth in Quadratic Number Fields for Elliptic Curves Defined Over the Rationals" by Arias-de-Reyna, Pineda-Martín, and Tornero.

1. Problem Statement

The paper addresses the inverse problem of torsion growth for elliptic curves.

• Context: It is well-established (via works like [3, 4, 11]) how the torsion subgroup $E_{tors}(K)$ of an elliptic curve $E/\mathbb{Q}$ can grow when the base field is extended to a quadratic number field $K = \mathbb{Q}(\sqrt{d})$.
• The Question: Given an elliptic curve $E/\mathbb{Q}$ and the knowledge that torsion grows (i.e., $E_{tors}(\mathbb{Q}) \neq E_{tors}(K)$), what can be deduced about the field $K$? Specifically, can the prime divisors of the square-free integer $d$ be characterized by invariants of the curve, such as its conductor $N_E$?
• Goal: To establish an explicit relationship between the primes dividing $d$ and the primes dividing $N_E$ (the conductor of $E$).

2. Methodology

The authors employ a combination of Galois representation theory, valuation theory, and explicit arithmetic geometry.

• Galois Representations: The core tool is the mod $\ell$ Galois representation $\rho_{E,\ell}: G_{\mathbb{Q}} \to \text{Aut}(E[\ell]) \simeq \text{GL}_2(\mathbb{F}_\ell)$. The existence of a new torsion point $P \in E(K) \setminus E(\mathbb{Q})$ implies that the quadratic field $K$ is contained in the extension $\mathbb{Q}(E[\ell])$.
• Subgroup Lattice Analysis: The authors analyze the lattice of subgroups of $\text{GL}_2(\mathbb{F}_\ell)$ for $\ell \in \{2, 3, 5, 7\}$. They determine which subgroups $H \subset \text{GL}_2(\mathbb{F}_\ell)$ (representing the image of the Galois group of $K$) possess fixed points (corresponding to torsion points) that are not fixed by the full Galois group of $\mathbb{Q}$.
• Valuation and Reduction Types:
  • They utilize the Néron–Ogg–Shafarevich Criterion to relate ramification in $\mathbb{Q}(E[\ell])$ to the primes dividing the conductor $N_E$ or the prime $\ell$.
  • They perform detailed 2-adic and 3-adic valuation analyses on the coefficients of minimal Weierstrass models and Tate normal forms.
  • They distinguish between strict growth (a new point of prime order appears) and mixed growth (a new point $P$ appears such that a multiple $mP$ is already in $E(\mathbb{Q})$).
• Case-by-Case Analysis: The paper treats the primes $\ell = 2, 3, 5, 7$ separately, as these are the only primes for which torsion growth can occur over quadratic fields.

3. Key Contributions and Results

A. The Prime $\ell = 2$ (Sections 2 & 3)

This is the most complex case, involving both strict and mixed growth scenarios.

• Theorem 2: If $E/\mathbb{Q}$ has torsion growth over a quadratic field $K = \mathbb{Q}(\sqrt{d})$ and $2 \mid d$, then **$2 \mid N_E$**.
  • Implication: If $E$ has good reduction at 2, torsion cannot grow over a field ramified at 2.
• Proof Strategy: The authors analyze the Galois group $\text{Gal}(\mathbb{Q}(E[2])/\mathbb{Q})$. They show that if $E$ has good reduction at 2, the structure of the 2-division polynomial and the discriminant leads to a contradiction regarding the 2-adic valuation of the discriminant if $2 \mid d$.
• Mixed Case: They prove that even in the "mixed" case (where a point of order 4, 8, or 16 appears such that its double is rational), the condition $2 \mid d$ forces bad reduction at 2.

B. The Prime $\ell = 3$ (Section 4)

The prime 3 behaves uniquely because it allows for "unexpected" growth (e.g., $C_1 \to C_3$) even when $3 \nmid N_E$(good reduction), provided the field is$\mathbb{Q}(\sqrt{-3})$.

• Proposition 7: If $E_{tors}(\mathbb{Q})$ is trivial and $E_{tors}(K) \cong C_9$, then $E$ must have bad reduction at 3.
• Proposition 8: If $E$ has good reduction at 3 and $3 \mid d$, and a new 3-torsion point appears, then the growth is strictly$E_{tors}(K) = E_{tors}(\mathbb{Q}) \times C_3$.
  • Significance: This characterizes the specific conditions under which 3 divides $d$ without dividing $N_E$.

C. The Primes $\ell = 5, 7$ (Section 5)

• Theorem 3: If $p \in \{5, 7\}$ ramifies in $K$ (i.e., $p \mid d$) and torsion grows, then $p \mid N_E$.
  • Methodology: The authors analyze the inertia group $I_\ell$ at $\ell$. They show that for $\ell=5,7$, the inertia group is too large to allow a quadratic subfield to fix a non-trivial torsion point unless the curve has bad reduction at $\ell$.
  • They utilize detailed lists of subgroups of $\text{GL}_2(\mathbb{F}_5)$ and $\text{GL}_2(\mathbb{F}_7)$ (referencing Sutherland and Zywina) to rule out the existence of index-2 subgroups with fixed points in the case of good reduction.

D. Main Theorems and Stronger Results

• Theorem 5 (Main Synthesis): Let $E/\mathbb{Q}$ have conductor $N_E$ and $K = \mathbb{Q}(\sqrt{d})$ be a quadratic field with torsion growth. If a prime $p$ divides $d$, then either $p \mid N_E$ or $p = 3$.
  • This provides a "sieve" for the possible quadratic fields: one only needs to check fields ramified at primes dividing the conductor or the prime 3.
• Theorem 4 (Stronger Result): The authors prove a stronger statement than Theorem 3. If $P \in E(K)[\ell] \setminus E(\mathbb{Q})$ for a prime $\ell \geq 3$:
  1. For any prime $p \neq \ell$ ramifying in $K$, $E$ has additive reduction at $p$ (i.e., $p^2 \mid N_E$).
  2. If $p = \ell > 3$ ramifies in $K$, $E$ has additive reduction at $p$ (i.e., $p^2 \mid N_E$).

4. Significance and Impact

• Inverse Problem Solution: The paper moves beyond the forward problem (predicting growth) to the inverse problem (deducing field properties from growth), providing a concrete necessary condition on the field discriminant.
• Refinement of Known Results: While the result for $p > 3$ was known via different techniques (twists and abelian varieties), the authors provide a new proof using mod $\ell$ Galois representations that yields the stronger Theorem 4 (additive reduction requirement).
• Detailed Classification of $\ell=2$: The paper offers a comprehensive analysis of the $\ell=2$ case, distinguishing between strict and mixed growth and proving that bad reduction at 2 is necessary for growth over fields ramified at 2.
• Characterization of $\ell=3$: It clarifies the exceptional nature of the prime 3, showing that while it can ramify without dividing the conductor, it imposes strict structural constraints on the torsion subgroup (e.g., ruling out $C_9$ growth with good reduction).

5. Conclusion

The authors successfully establish that the set of quadratic fields where torsion growth occurs is highly restricted. Specifically, the primes ramifying in such fields must divide the conductor of the curve (implying bad reduction), with the sole exception of the prime 3, which requires specific conditions on the torsion structure. This work lays the groundwork for a complete classification of quadratic fields supporting torsion growth based on the arithmetic invariants of the elliptic curve.