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On the generalized Fermat equation of signature $(5,p,3)$

Ariel Pacetti, Lucas Villagra Torcomian

TL;DR

The paper introduces hypergeometric motives as a new conduit for the modular method applied to the generalized Fermat equation with signature (5,p,3). By constructing explicit rank-2 motives and analyzing their conductors, traces, and residual images, the authors classify the possible Galois-automorphic realizations of a putative solution and isolate obstructions from trivial, CM, and ghost cases. Conditional on a standard large-image conjecture, they prove nonexistence of nontrivial primitive solutions with 3 ∤ c for p sufficiently large, and for p outside a finite set they obtain unconditional eliminations via Mazur-type congruences, yielding the first Diophantine result for an infinite family of signatures with three distinct prime exponents. The work also maps out a detailed program to manage ghost obstructions and provides explicit hyperelliptic realizations and Hilbert modular-form data in the (5,p,3) case, illustrating the practical viability of hypergeometric-motives in modular-method arguments.

Abstract

In this article we study solutions to the generalized Fermat equation $x^q+y^p+z^r=0 $ using hypergeometric motives within the framework of the modular method. In doing so, we give an explicit description of the ramification behavior at primes dividing $2qr$ and analyze the contribution of trivial solutions. We identify a general obstruction to the modular method that accounts for its failure in many instances. As an application, assuming a standard large image conjecture, we prove that the previous equation admits no nontrivial primitive solutions $(a,b,c)$ with $3 \nmid c$, when $q=5,$ $r=3$ and $p$ is a prime sufficiently large.

On the generalized Fermat equation of signature $(5,p,3)$

TL;DR

The paper introduces hypergeometric motives as a new conduit for the modular method applied to the generalized Fermat equation with signature (5,p,3). By constructing explicit rank-2 motives and analyzing their conductors, traces, and residual images, the authors classify the possible Galois-automorphic realizations of a putative solution and isolate obstructions from trivial, CM, and ghost cases. Conditional on a standard large-image conjecture, they prove nonexistence of nontrivial primitive solutions with 3 ∤ c for p sufficiently large, and for p outside a finite set they obtain unconditional eliminations via Mazur-type congruences, yielding the first Diophantine result for an infinite family of signatures with three distinct prime exponents. The work also maps out a detailed program to manage ghost obstructions and provides explicit hyperelliptic realizations and Hilbert modular-form data in the (5,p,3) case, illustrating the practical viability of hypergeometric-motives in modular-method arguments.

Abstract

In this article we study solutions to the generalized Fermat equation using hypergeometric motives within the framework of the modular method. In doing so, we give an explicit description of the ramification behavior at primes dividing and analyze the contribution of trivial solutions. We identify a general obstruction to the modular method that accounts for its failure in many instances. As an application, assuming a standard large image conjecture, we prove that the previous equation admits no nontrivial primitive solutions with , when and is a prime sufficiently large.

Paper Structure

This paper contains 38 sections, 40 theorems, 121 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Hypergeometric motives: a new approach
  3. A general obstruction
  4. Signature $(5,p,3)$
  5. Hypergeometric motives
  6. Geometric properties
  7. Construction of the motive
  8. Specializations
  9. Traces of Frobenius elements
  10. Conductor at potentially wild primes
  11. Local inertial type
  12. Conductor at primes dividing $r$
  13. The case $r=3$
  14. The case $r\ge5$
  15. Irreducibility of $F(x)$
  16. ...and 23 more sections

Key Result

Theorem 1

Attached to a hypothetical solution $(a,b,c)$ to there exist two Galois representations where $K={\mathbb Q}(\zeta_q)^+\cdot{\mathbb Q}(\zeta_r)^+$ and ${\mathfrak{p}}$ is a prime in $K$ above $p$, satisfying the following. If $\bar{\rho}^\pm_{\mathfrak{p}}$ is irreducible, then: The conductor exponent at primes dividing $qr$ is described in Corollary coro:conductor-at-r, Proposition prop:condu

Theorems & Definitions (101)

  • Theorem 1
  • Theorem A
  • Theorem B
  • Conjecture 1
  • Theorem C
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • ...and 91 more