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A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves

Imin Chen, Angelos Koutsianas

TL;DR

The paper develops a modular-Darmon framework for the generalized Fermat equation $x^n+y^n=z^5$ with signature $(p,p,5)$, using Frey hyperelliptic curves $J_5^\ obracket{\pm}$ of dimension $>1$. It establishes modularity (notably for $J^-$) and computes precise conductors and Serre levels for residual representations, combining local ramification analyses with level-lowering arguments. By controlling irreducibility and employing twists to minimize ramification, the authors eliminate large classes of nontrivial primitive solutions and reduce the problem to a big image or CM-form obstruction, yielding optimal bounds on the exponent and clarifying the obstructions that remain. The work also contributes novel conductor computations at primes above $2$ and $5$ for genus $2$ Frey curves, and situates the results within Darmon–Duke’s program, including refinements to their conductor- and twist-based strategy. Overall, this advances the modular approach to higher-dimensional Frey varieties in the generalized Fermat landscape and delineates the precise arithmetic obstacles that persist in signature $(p,p,5)$ cases.

Abstract

In this paper we carry out the steps of Darmon's program for the generalized Fermat equation $$ x^n + y^n = z^5. $$ In particular, we develop the machinery necessary to prove an optimal bound on the exponent $n$ for solutions satisfying certain $2$-adic and $5$-adic conditions which are natural from the point of view of the method. We also reduce the problem of resolving this equation to a `big image conjecture', completing a line of ideas suggested in his original program. The above equation is an example of a generalized Fermat equation for which the predicted Frey abelian varieties have dimension $ > 1$ and thus it represents an interesting test case for Darmon's program.

A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves

TL;DR

The paper develops a modular-Darmon framework for the generalized Fermat equation with signature , using Frey hyperelliptic curves of dimension . It establishes modularity (notably for ) and computes precise conductors and Serre levels for residual representations, combining local ramification analyses with level-lowering arguments. By controlling irreducibility and employing twists to minimize ramification, the authors eliminate large classes of nontrivial primitive solutions and reduce the problem to a big image or CM-form obstruction, yielding optimal bounds on the exponent and clarifying the obstructions that remain. The work also contributes novel conductor computations at primes above and for genus Frey curves, and situates the results within Darmon–Duke’s program, including refinements to their conductor- and twist-based strategy. Overall, this advances the modular approach to higher-dimensional Frey varieties in the generalized Fermat landscape and delineates the precise arithmetic obstacles that persist in signature cases.

Abstract

In this paper we carry out the steps of Darmon's program for the generalized Fermat equation In particular, we develop the machinery necessary to prove an optimal bound on the exponent for solutions satisfying certain -adic and -adic conditions which are natural from the point of view of the method. We also reduce the problem of resolving this equation to a `big image conjecture', completing a line of ideas suggested in his original program. The above equation is an example of a generalized Fermat equation for which the predicted Frey abelian varieties have dimension and thus it represents an interesting test case for Darmon's program.
Paper Structure (18 sections, 37 theorems, 111 equations)

This paper contains 18 sections, 37 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Hyperelliptic equations
  4. Abelian varieties
  5. Darmon's Frey hyperelliptic curve for signature $(p,p,r)$
  6. Modularity of $\rho_{J^\pm,\mathfrak{p}}$
  7. Conductors of $\bar{\rho}_{J^{\pm},\mathfrak{p}}$
  8. The conductor of $(\rho_{J^+,\lambda})$
  9. The conductor of $(\rho_{J^-,\lambda})$
  10. The conductor of $(\rho_{J^-,\lambda})$ at $\mathfrak{q}_2$
  11. The conductor of $(\rho_{J^-,\lambda})$ at $\mathfrak{q}_5$
  12. Proof of Theorem \ref{['thm:Serre_level']}
  13. Irreducibility of $\bar{\rho}_{J^{\pm},\mathfrak{p}}$
  14. Proof of Theorem \ref{['main-theorem']}
  15. Local comparison of traces
  16. ...and 3 more sections

Key Result

Theorem 1.1

For $n \ge 3$, there are no non-trivial primitive solutions $(a,b,c) \in \mathbb{Z}^3$ to the equation in the cases

Theorems & Definitions (89)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6
  • proof
  • ...and 79 more