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Asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields

Satyabrat Sahoo, Narasimha Kumar

Abstract

In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields $K$, i.e., $Ax^p+By^p=Cz^3$ with prime exponent $p$ and $A,B,C \in \mathcal{O}_K \setminus \{0\}$. For certain class of fields $K$, we prove that $Ax^p+By^p=Cz^3$ has no asymptotic solutions over $K$ (resp., solutions of certain type over $K$) with restrictions on $A,B,C$ (resp., for all $A,B,C \in \mathcal{O}_K \setminus \{0\}$). Finally, we present several local criteria over $K$.

Asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields

Abstract

In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature over totally real number fields , i.e., with prime exponent and . For certain class of fields , we prove that has no asymptotic solutions over (resp., solutions of certain type over ) with restrictions on (resp., for all ). Finally, we present several local criteria over .
Paper Structure (18 sections, 17 theorems, 14 equations, 1 table)

This paper contains 18 sections, 17 theorems, 14 equations, 1 table.

Table of Contents

  1. Introduction
  2. Literature Survey:
  3. Literature for the equations of signature $(p,p,p)$ and $(p,p,2)$
  4. Preliminaries
  5. Our strategy:
  6. Solutions of the equation $Ax^p+By^p=Cz^3$ over $W_K$
  7. Main result
  8. Construction of Frey elliptic curves
  9. Modularity of Frey elliptic curves
  10. Reduction type of Frey elliptic curves
  11. Type of reduction with image of inertia
  12. Proof of Theorem \ref{['main result1 for x^2=By^p+Cz^p Type I']}.
  13. Solutions of $Ax^p+By^p=Cz^3$ over $K$
  14. Main result
  15. Modularity of Frey elliptic curves
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $E$ be an elliptic curve over $K$ of conductor $\mathfrak{n}$. Let $p$ be a rational prime. Suppose that the following conditions hold: Then there exists a Hilbert modular newform $f$ over $K$ of parallel weight $2$, level $\mathfrak{n}_p$, and some prime $\omega$ of $\mathbb{Q}_f$ such that $\omega | p$ and $\bar{\rho}_{E,p} \sim \bar{\rho}_{f,\omega}$.

Theorems & Definitions (38)

  • Conjecture 1.1: Eichler-Shimura
  • Theorem 1.2
  • Definition 2.1: Trivial solution
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 2.7
  • Theorem 2.8
  • ...and 28 more