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Asymptotic Fermat equation of signature $(r, r, p)$ over totally real fields

Somnath Jha, Satyabrat Sahoo

TL;DR

The paper addresses asymptotic solutions of the Fermat equation with signature $(r,r,p)$ over totally real fields by deploying the modular method with Frey curves over $K^+$ and $K$. It establishes two main results: (i) a general nonexistence result for large $p$ under explicit $S$-unit bound hypotheses, and (ii) a specialized result for $x^5+y^5=dz^p$ ruling out large-$p$ solutions within a restricted set $W_K$, together with local criteria and corollaries. The approach combines modularity and level-lowering for totally real fields, detailed conductor and reduction analyses at primes above $2$, and explicit connections to $S$-unit equations, yielding computable bounds $V$ and verifiable local conditions. These contributions extend asymptotic Fermat-type results to broader number-field settings, providing concrete criteria to certify nonexistence of asymptotic solutions and illustrating the power of the Frey-curve/Modularity framework in arithmetic geometry.

Abstract

Let $K$ be a totally real number field and $ \mathcal{O}_K$ be the ring of integers of $K$. This manuscript examines the asymptotic solutions of the Fermat equation of signature $(r, r, p)$, specifically $x^r+y^r=dz^p$ over $K$, where $r,p \geq5$ are rational primes and odd $d\in \mathcal{O}_K \setminus \{0\}$. For a certain class of fields $K$, we first prove that the equation $x^r+y^r=dz^p$ has no asymptotic solution $(a,b,c) \in \mathcal{O}_K^3$ with $2 |c$. Then, we study the asymptotic solutions $(a,b,c) \in \mathcal{O}_K^3$ to the equation $x^5+y^5=dz^p$ with $2 \nmid c$. We use the modular method to prove these results.

Asymptotic Fermat equation of signature $(r, r, p)$ over totally real fields

TL;DR

The paper addresses asymptotic solutions of the Fermat equation with signature over totally real fields by deploying the modular method with Frey curves over and . It establishes two main results: (i) a general nonexistence result for large under explicit -unit bound hypotheses, and (ii) a specialized result for ruling out large- solutions within a restricted set , together with local criteria and corollaries. The approach combines modularity and level-lowering for totally real fields, detailed conductor and reduction analyses at primes above , and explicit connections to -unit equations, yielding computable bounds and verifiable local conditions. These contributions extend asymptotic Fermat-type results to broader number-field settings, providing concrete criteria to certify nonexistence of asymptotic solutions and illustrating the power of the Frey-curve/Modularity framework in arithmetic geometry.

Abstract

Let be a totally real number field and be the ring of integers of . This manuscript examines the asymptotic solutions of the Fermat equation of signature , specifically over , where are rational primes and odd . For a certain class of fields , we first prove that the equation has no asymptotic solution with . Then, we study the asymptotic solutions to the equation with . We use the modular method to prove these results.

Paper Structure

This paper contains 23 sections, 30 theorems, 30 equations, 1 table.

Table of Contents

  1. Introduction
  2. Structure of the article
  3. Statement of the main results
  4. Main result for the solutions of $x^r+y^r=dz^p$ over $K$
  5. Main result for the solutions of $x^5+y^5=dz^p$ over $K$
  6. Comparison with the works of M23, KMO24:
  7. Proof of Theorem \ref{['main result1 for (r,r,p)']}
  8. Construction of Frey elliptic curve
  9. Conductor of the Frey elliptic curve
  10. Modularity of the Frey elliptic curve
  11. Irreducibility of the mod $p$ Galois representations attached to elliptic curves
  12. Level lowering results
  13. Eichler-Shimura
  14. Reduction type of the Frey elliptic curve at primes lying above $2$
  15. An auxiliary result for Theorem \ref{['main result1 for (r,r,p)']}
  16. ...and 8 more sections

Key Result

Theorem 2.2

Let $K$ be a totally real number field, $r\geq 5$ be a fixed rational prime and $d\in \mathcal{O}_K \setminus \{0\}$ be odd. Let $K^+:=K(\zeta_r+ \zeta_r^{-1})$. Suppose, for every solution $(\lambda, \mu)$ to the $S_{K^+, 2rd}$-unit equation there exists a prime $\mathfrak{P} \in S_{{K^+}, 2}$ that satisfies Then, there exists a constant $V=V_{K,r,d}>0$ (depending on $K,r,d$) such that for prim

Theorems & Definitions (61)

  • Conjecture 1.1
  • Definition 2.1
  • Theorem 2.2: Main result 1
  • Remark 2.3
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8: Main result 2
  • ...and 51 more