🔢 mathematics

Non-trivial Solutions of $Aa^p+Bb^p=Cc^3$ over Number Fields

This paper employs the modular method to establish both asymptotic and effective results for the Diophantine equation $Aa^p + Bb^p = Cc^3$ over number fields, including explicit bounds on the exponent $p$ that guarantee the non-existence of certain non-trivial solutions in specific imaginary quadratic fields.

Original authors: Yasemin Kara, Stef Nomden, Ekin Özman

Published 2026-04-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yasemin Kara, Stef Nomden, Ekin Özman

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 very specific, ancient mystery: The Case of the Missing Numbers.

For centuries, mathematicians have been trying to figure out if you can find three whole numbers ( $a, b, c$ ) that fit into a very tricky equation:
$a^p + b^p = c^3$
(Here, $p$ is a special kind of number called a "prime," like 2, 3, 5, 7, 11, etc., and we are looking for solutions where none of the numbers are zero).

This paper, written by Yasemin Kara, Stef Nomden, and Ekin Özman, is about how they used a high-tech detective tool called the "Modular Method" to prove that for very large prime numbers ( $p$ ), no such solutions exist in certain mathematical worlds called "Number Fields."

Here is the breakdown of their investigation, explained with simple analogies.

1. The Detective's Toolkit: The Modular Method

To solve this, the authors didn't just try plugging in numbers (which would take forever). Instead, they used a strategy invented to solve Fermat's Last Theorem (the most famous math problem of the 20th century).

Think of the Modular Method as a "Shadow Casting" technique.

The Suspect: If a solution to the equation did exist, the authors imagine building a special "Frey Curve" (a specific type of geometric shape, like a donut with a hole).
The Shadow: Every geometric shape casts a shadow. In math, this shadow is a "Galois Representation." It's a code that describes the shape's properties.
The Alibi: The authors check if this shadow matches any known "Modular Forms" (which are like a library of pre-approved, perfect shadows).

The Logic: If the suspect (the solution) exists, it must cast a shadow that matches a library entry. But the authors prove that for large $p$ , the shadow cast by the suspect is weird and impossible. It doesn't match anything in the library. Therefore, the suspect (the solution) never existed.

2. The Setting: Number Fields

Most people know about regular numbers ( $\mathbb{Q}$ ), but this paper investigates "Number Fields."

Analogy: Imagine regular numbers are a flat, 2D map. Number fields are like 3D landscapes or different dimensions.
The authors focus on Imaginary Quadratic Fields. Think of these as landscapes where you can take a step "sideways" into a world of imaginary numbers (like $\sqrt{-7}$ or $\sqrt{-19}$ ).
They ask: "If we live in these weird 3D landscapes, can we still find solutions to our equation?"

3. The Two Main Discoveries

The paper presents two types of victories:

A. The "Asymptotic" Victory (The Big Picture)

The Result: They proved that if the prime number $p$ is big enough, there are no solutions in these landscapes, provided the landscape meets certain "S-unit" conditions.

The Analogy: Imagine you are looking for a specific type of bird in a forest. You can't check every single tree. But you prove that if the forest is big enough and has a certain type of tree, no matter how far you walk, you will never find that bird.
They showed that for many imaginary quadratic fields, the "forest conditions" are right, so for huge primes, the equation is impossible.

B. The "Effective" Victory (The Specific Numbers)

The Result: They didn't just say "it's impossible for big numbers." They gave exact speed limits.

They looked at four specific landscapes: $\mathbb{Q}(\sqrt{-7})$ , $\mathbb{Q}(\sqrt{-19})$ , $\mathbb{Q}(\sqrt{-43})$ , and $\mathbb{Q}(\sqrt{-67})$ .
They calculated a specific "Cut-off Point" ( $B_K$ $B_{K}$ ) for each.
- For $\sqrt{-7}$ and $\sqrt{-19}$ : If $p > 20$ , no solution exists.
- For $\sqrt{-43}$ : If $p > 2,531$ , no solution exists.
- For $\sqrt{-67}$ : If $p > 86,338,229$ , no solution exists.
The Analogy: It's like a bouncer at a club. "If you are older than 20 (or 86 million), you cannot get in." They didn't just say "no kids allowed"; they gave the exact age limit for each specific club.

4. The Hurdles and Assumptions

The authors had to make a few assumptions because math is still a work in progress.

The "Conjecture" Hurdle: To use their "Shadow Casting" tool, they had to assume that certain mathematical bridges (called Modularity Conjectures) exist. These bridges connect the geometric shapes to the library of shadows.
The Analogy: It's like saying, "If we assume the bridge to the library is sturdy, then we can prove the bird doesn't exist." The authors believe the bridge is sturdy, but they can't prove the bridge itself yet.

5. How They Did the Math (The "Inertia Argument")

For the specific numbers (like 7, 19, 43, 67), they had to do some heavy lifting with computers (using software called Magma).

They checked the "library" of shadows to see if any matched the weird shadow cast by the equation.
They found that for these specific numbers, the library was empty for the required conditions.
The "Inertia" Trick: In one case, they used a clever trick involving how the shape behaves when you squeeze it (mathematically called "reduction"). They showed the shape would have to behave in two contradictory ways at once, which is impossible.

Summary

This paper is a triumph of modern detective work in mathematics.

The Crime: Finding solutions to $a^p + b^p = c^3$ in complex number worlds.
The Method: Using "Shadows" (Galois representations) to prove the solution is a fake.
The Outcome: They proved that for huge prime numbers, the solution is impossible.
The Bonus: They gave exact "cut-off" numbers for four specific imaginary worlds, telling us exactly how big the prime needs to be to guarantee no solution exists.

It's a reminder that even in the abstract world of imaginary numbers, there are strict rules that prevent certain things from ever happening, no matter how hard you try to find them.

1. Problem Statement

The paper investigates the Diophantine equation:
$Aa^p + Bb^p = Cc^3$
over a general number field $K$ , where $A, B, C \in \mathcal{O}_K$ are non-zero coefficients, $p$ is a prime exponent, and $(a,b,c) \in \mathcal{O}_K^3$ is a primitive solution (the ideal generated by $a, b, c$ is the unit ideal).

The authors aim to prove two types of results:

Asymptotic Results: Showing that for a fixed number field $K$ and coefficients $A, B, C$ , there exists a constant $V$ such that no non-trivial primitive solutions exist for any prime $p > V$ .
Effective Results: Determining explicit bounds for $p$ for specific imaginary quadratic fields $K = \mathbb{Q}(\sqrt{-d})$ with specific coefficients ( $A=1, B=d, C=1$ ).

2. Methodology: The Modular Method

The paper employs the Modular Method, a strategy pioneered by Wiles, Ribet, and others to solve Fermat's Last Theorem. The general workflow involves:

Frey Curve Construction: Assuming a non-trivial primitive solution $(a,b,c)$ exists, the authors construct a specific elliptic curve (the Frey curve) $E_{a,b,c}$ over $K$ :
$Y^2 + 3CcXY + C^2Bb^pY = X^3$
This curve has a rational point of order 3 and specific arithmetic invariants ( $\Delta, j, c_4, c_6$ ) dependent on the solution.
Galois Representations: They analyze the mod $p$ Galois representation $\rho_{E,p}: G_K \to \text{GL}_2(\mathbb{F}_p)$ associated with the $p$ -torsion points of $E$ .
- Irreducibility: They prove that for sufficiently large $p$ , this representation is absolutely irreducible.
- Semistability: They establish that $E$ is semistable outside a finite set of primes $T_K$ (primes dividing $3ABC$).
Modularity and Level Lowering:
- Conjectures: The method relies on two unproven conjectures for general number fields:
  - Conjecture 2.1: Modularity of odd, irreducible, continuous representations (Serre-type).
  - Conjecture 2.2: The Langlands correspondence linking eigenforms to elliptic curves (or fake elliptic curves).
- Level Lowering: Assuming modularity, the representation $\rho_{E,p}$ is shown to arise from a modular form $\theta$ of level $N_E$ (the Serre conductor). Using level lowering techniques, they construct a new elliptic curve $E'$ (or a fake elliptic curve) with a lower level $N'$ that is independent of the specific solution $(a,b,c)$ but depends only on $K, A, B, C$ .
Contradiction via S-Unit Equations:
- The curve $E'$ must have a $K$ -rational point of order 3. This imposes strict constraints on its $j$ -invariant.
- The condition that $E'$ has good reduction outside $T_K$ forces the $j$ -invariant to be an $S$ -unit (where $S = T_K$ ).
- This leads to an equation of the form $\alpha + \beta = \gamma^3$ in $S$ -units.
- The authors analyze the valuations of these units at primes above 3. If the valuations satisfy certain bounds (derived from the properties of the Frey curve), a contradiction arises because the $j$ -invariant of $E'$ would have to have negative valuation at a prime where it must have non-negative valuation (due to good reduction).

3. Key Contributions

A. Generalization to Imaginary Quadratic Fields

Previous work (e.g., Freitas-Siksek, Deconinck) largely focused on totally real number fields. This paper extends the modular method to general number fields, specifically handling imaginary quadratic fields.

Technical Innovation: The authors use a modified level-lowering technique (inspired by Isik, Kara, and Özman) that allows for this generality.
Trade-off: While totally real fields often require only one modularity conjecture, the general case requires assuming two conjectures (Conjectures 2.1 and 2.2).

B. Asymptotic Theorems with S-Unit Conditions

The paper establishes Theorem 1.1, which provides an asymptotic result for general number fields $K$ satisfying:

$Cl_{T_K}(K)[3] = 1$ (No 3-torsion in the $T_K$ -class group).
A specific condition on solutions to the $S$ -unit equation $\alpha + \beta = \gamma^3$ : For every solution, there exists a prime $L$ above 3 such that $|v_L(\alpha\beta^{-1})| \leq 3v_L(3)$ .

This condition is verified for several imaginary quadratic fields in Corollary 1.3.

C. Effective Bounds for Specific Fields

The paper provides Theorem 1.4, giving explicit bounds for the equation $a^p + db^p = c^3$ over $K = \mathbb{Q}(\sqrt{-d})$ for $d \in \{7, 19, 43, 67\}$ .

The authors compute the torsion subgroups of the abelianization of congruence subgroups $\Gamma_0(N_E)$ to determine the explicit bound $B(N)$ required to lift mod $p$ eigenforms to complex eigenforms.
They use computational tools (Magma) to eliminate potential modular forms that could correspond to a solution.

4. Main Results

Asymptotic Results

Theorem 1.1: Under Conjectures 2.1 and 2.2, and specific $S$ -unit conditions, the equation $Aa^p + Bb^p = Cc^3$ has no non-trivial primitive solutions for $p > V(K, A, B, C)$ .
Corollary 1.3: For $K = \mathbb{Q}(\sqrt{-d})$ with $d \equiv 1 \pmod 3$ , if $3 \nmid h_K h_K(\zeta_3)$ and the field satisfies the conjectures, the equation has no solutions for large $p$ .

Effective Results

For the specific equation $a^p + db^p = c^3$ over $K = \mathbb{Q}(\sqrt{-d})$ with $d \in \{7, 19, 43, 67\}$ , assuming Conjecture 2.1:

$d=7, 19$ : No non-trivial solutions exist for $p > 20$ .
$d=43$ : No non-trivial solutions exist for $p > 2531$ .
$d=67$ : No non-trivial solutions exist for $p > 86,338,229$ .

Note: These bounds apply to solutions where $(a, db, c)$ is primitive and the prime $\lambda$ above 3 divides $b$ . In some cases (Remark 1.5), the condition on the type of solution can be relaxed, but the proofs are omitted due to length.

5. Significance

Extension of the Modular Method: This work is a significant step in applying the modular method beyond totally real fields to complex quadratic fields, which present additional challenges regarding the lack of real embeddings and the nature of the associated Galois representations.
Explicit Computation: The paper moves beyond theoretical existence to provide explicit numerical bounds for specific fields, bridging the gap between abstract number theory and computational verification.
Handling the Exponent 3: The equation involves a cubic term ( $c^3$ ) rather than the symmetric $c^p$ . The authors successfully adapt the Frey curve construction and level-lowering arguments to handle this mixed signature $(p, p, 3)$ , which requires careful analysis of the 3-torsion structure and the behavior of the curve at primes above 3.
Verification of Conjectures: By showing that the necessary $S$ -unit conditions hold for specific imaginary quadratic fields, the paper demonstrates that the theoretical framework is applicable to concrete cases, provided the modularity conjectures hold.

In summary, the paper successfully adapts the modular method to prove the non-existence of solutions for a generalized Fermat-type equation with a cubic term over imaginary quadratic fields, providing both general asymptotic theorems and specific effective bounds.