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Solving equations of signature $(p,p,2)$ with coefficients over number fields

Begum Gulsah Cakti, Erman Isik, Yasemin Kara, Ekin Ozman

Abstract

Using the modular method, we study solutions to the Diophantine equation $$Aa^p+Bb^p=Cc^2$$ over number fields. We first prove an asymptotic result for general number fields satisfying an appropriate $S$-unit condition by assuming some standard conjectures in the case of fields that are not totally real. Specifically, we verify that this condition holds for an infinite family of real quadratic fields. Outside the asymptotic setting, we also obtain effective results. In particular, for the equation $$a^p+db^p=c^2$$ over $K= \mathbb{Q}(\sqrt{-d})$ with $d \in \{3, 11, 19, 43, \}$ and $K= \mathbb{Q}(\sqrt d)$ with $d \in \{3, 5, 11, 13, 19, 29\}$, we find explicit bounds (depending on $d$) such that no non-trivial solutions of a certain type exist whenever $p$ exceeds these bounds.

Solving equations of signature $(p,p,2)$ with coefficients over number fields

Abstract

Using the modular method, we study solutions to the Diophantine equation over number fields. We first prove an asymptotic result for general number fields satisfying an appropriate -unit condition by assuming some standard conjectures in the case of fields that are not totally real. Specifically, we verify that this condition holds for an infinite family of real quadratic fields. Outside the asymptotic setting, we also obtain effective results. In particular, for the equation over with and with , we find explicit bounds (depending on ) such that no non-trivial solutions of a certain type exist whenever exceeds these bounds.
Paper Structure (22 sections, 36 theorems, 64 equations)

This paper contains 22 sections, 36 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Frey Curve and Related Facts
  3. Conductor of the Frey Curve
  4. Image of Inertia
  5. Modularity of the Frey Curve
  6. Totally real case
  7. General case
  8. Properties of Galois Representations and Level Lowering
  9. Irreducibility of the associated Galois representation $\overline{\rho}_{E,p}$
  10. Level Lowering and Eichler-Shimura
  11. Totally real case
  12. General case
  13. Asymptotic Results
  14. Proof of Theorem \ref{['maintheoremA']}
  15. The proof of Theorem \ref{['maintheoremB']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $K$ be a number field with $\mathop{\mathrm{Cl}}\nolimits_{S_K}(K)[2]=\{1\}$. In the case where $K$ is not totally real, we assume both Conjecture conj1 and Conjecture conj2 hold for $K$. Suppose that there exists some distinguished prime $\widetilde{{\mathfrak P}}\in T_{K}$ such that every solu satisfies $|v_{\widetilde{{\mathfrak P}}}(\alpha/\beta)|\leq 6v_{\widetilde{{\mathfrak P}}}(2)$. Th

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2: FS, Lemma 3.4
  • ...and 58 more