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

Yasemin Kara, Stef Nomden, Ekin Özman

TL;DR

The paper extends the modular method to the generalized Fermat equation $Aa^p+Bb^p=Cc^3$ over number fields, under standard modularity conjectures. It develops an asymptotic nonexistence result for primitive nontrivial solutions by leveraging a Frey curve, level lowering, and properties of Galois representations, together with an $S$-unit condition. In addition, it provides an explicit effective bound for imaginary quadratic fields $K=Q( ext{-}d)$ with $d\in\\{7,19,43,67\\}$, giving concrete $p$-thresholds beyond which no solutions of a specified type exist; these bounds are obtained via lifting mod-$p$ eigenforms to complex ones and performing finite computations using Magma and modular form databases. The results generalize prior work on totally real fields and imaginary quadratic cases, offering a framework to obtain both asymptotic and explicit nonexistence results for Diophantine equations over number fields within the modular paradigm. This advances understanding of how modularity, $S$-unit conditions, and level-lowering interact to constrain high-exponent solutions in arithmetic geometry.

Abstract

In this paper, we investigate solutions to the Diophantine equation $ A a^p + B b^p = C c^3 $ over number fields using the modular method. Assuming certain standard modularity conjectures, we first establish an asymptotic result for general number fields satisfying an appropriate $S$-unit condition. In particular, we verify that this condition holds for several imaginary quadratic fields. Beyond the asymptotic setting, we also obtain an effective result. Specifically, for the equation $a^p + d b^p = c^3$ over $ K = \mathbb{Q}(\sqrt{-d}) $ with $ d \in \{7, 19, 43, 67\} $, we determine an explicit bound (depending on $ d $) such that no solutions of a certain type exist whenever $ p $ exceeds this bound.

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

TL;DR

The paper extends the modular method to the generalized Fermat equation over number fields, under standard modularity conjectures. It develops an asymptotic nonexistence result for primitive nontrivial solutions by leveraging a Frey curve, level lowering, and properties of Galois representations, together with an -unit condition. In addition, it provides an explicit effective bound for imaginary quadratic fields with , giving concrete -thresholds beyond which no solutions of a specified type exist; these bounds are obtained via lifting mod- eigenforms to complex ones and performing finite computations using Magma and modular form databases. The results generalize prior work on totally real fields and imaginary quadratic cases, offering a framework to obtain both asymptotic and explicit nonexistence results for Diophantine equations over number fields within the modular paradigm. This advances understanding of how modularity, -unit conditions, and level-lowering interact to constrain high-exponent solutions in arithmetic geometry.

Abstract

In this paper, we investigate solutions to the Diophantine equation over number fields using the modular method. Assuming certain standard modularity conjectures, we first establish an asymptotic result for general number fields satisfying an appropriate -unit condition. In particular, we verify that this condition holds for several imaginary quadratic fields. Beyond the asymptotic setting, we also obtain an effective result. Specifically, for the equation over with , we determine an explicit bound (depending on ) such that no solutions of a certain type exist whenever exceeds this bound.
Paper Structure (12 sections, 21 theorems, 55 equations)

This paper contains 12 sections, 21 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conjectures
  4. Frey curve and related facts
  5. Properties of Galois representations
  6. Level lowering
  7. Irreducibility and absolute irreducibility of Galois Representations
  8. Asymptotic Results
  9. Proof of Theorem \ref{['main theorem']}
  10. Proof of Theorem \ref{['theorem: special case']}
  11. Quadratic imaginary number fields and proof of Corollary \ref{['corollary: quadratic imaginary fields asymptotic solutions']}
  12. Effective Result

Key Result

Theorem 1.1

Let $K$ be a number field with $\textup{Cl}_{T_K}(K)[3]=1$ satisfying Conjectures conj1 and conj2. Further suppose that for every solution $(\alpha,\beta,\gamma) \in\mathcal{O}_{T_K}^\times\times\mathcal{O}_{T_K}^\times\times\mathcal{O}_{T_K}$ to $\alpha+\beta=\gamma^3$, there exists a prime $\mathf Let $W_K$ be the set $(a,b,c)\in\mathcal{O}_K^3$ such that $(a,b,c)$ is a primitive non-trivial sol

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Conjecture 2.1: FKS, Conjecture 4.1
  • Conjecture 2.2: SS, Conjecture 4.1
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • ...and 28 more