Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Darmon's program for the Generalized Fermat equation, II

Nicolas Billerey, Imin Chen, Luis Dielefait, Nuno Freitas

TL;DR

This work advances Darmon's program for the generalized Fermat equation by combining multi-Frey strategies with Frey abelian varieties to solve x^7 + y^7 = 3 z^n for all n ≥ 2 and to approach x^7 + y^7 = z^n via Cartan-big-image considerations. It develops two complementary routes: (i) twists of Frey curves over totally real fields and (ii) higher-dimensional Frey abelian varieties, notably Kraus’ hyperelliptic curve and its Jacobian J/K, to constrain residual representations and perform Hilbert modular form level-lowering eliminations. The paper achieves complete nonexistence results for the first equation and conditional (Cartan-based) progress toward the second, while also demonstrating substantial computational gains from employing Frey varieties, including faster eliminations and favorable conductor reductions. Overall, it strengthens the practical viability of Darmon’s modular-method program in higher-dimensional settings and clarifies how Cartan-type obstructions dictate the landscape of possible solutions, with direct implications for large-prime behavior and Diophantine applications. The results illustrate a concrete path toward resolving broad classes of generalized Fermat equations by exploiting the richer arithmetic of Frey varieties alongside traditional Frey curves.

Abstract

We obtain additional Diophantine applications of the methods surrounding Darmon's program for the generalized Fermat equation developed in the first part of this series of papers. As a first application, we use a multi-Frey approach combining two Frey elliptic curves over totally real fields, a Frey hyperelliptic curve over $\mathbb{Q}$ due to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equation $$x^7 + y^7 = 3 z^n$$ for all integers $n \ge 2$. Moreover, we explain how the use of higher dimensional Frey abelian varieties allows a more efficient proof of this result due to additional structures that they afford, compared to using only Frey elliptic curves. As a second application, we use some of these additional structures that Frey abelian varieties possess to show that a full resolution of the generalized Fermat equation $x^7 + y^7 = z^n$ depends only on the Cartan case of Darmon's big image conjecture. In the process, we solve the previous equation for solutions $(a,b,c)$ such that $a$ and $b$ satisfy certain $2$ or $7$-adic conditions and all $n \ge 2$.

On Darmon's program for the Generalized Fermat equation, II

TL;DR

This work advances Darmon's program for the generalized Fermat equation by combining multi-Frey strategies with Frey abelian varieties to solve x^7 + y^7 = 3 z^n for all n ≥ 2 and to approach x^7 + y^7 = z^n via Cartan-big-image considerations. It develops two complementary routes: (i) twists of Frey curves over totally real fields and (ii) higher-dimensional Frey abelian varieties, notably Kraus’ hyperelliptic curve and its Jacobian J/K, to constrain residual representations and perform Hilbert modular form level-lowering eliminations. The paper achieves complete nonexistence results for the first equation and conditional (Cartan-based) progress toward the second, while also demonstrating substantial computational gains from employing Frey varieties, including faster eliminations and favorable conductor reductions. Overall, it strengthens the practical viability of Darmon’s modular-method program in higher-dimensional settings and clarifies how Cartan-type obstructions dictate the landscape of possible solutions, with direct implications for large-prime behavior and Diophantine applications. The results illustrate a concrete path toward resolving broad classes of generalized Fermat equations by exploiting the richer arithmetic of Frey varieties alongside traditional Frey curves.

Abstract

We obtain additional Diophantine applications of the methods surrounding Darmon's program for the generalized Fermat equation developed in the first part of this series of papers. As a first application, we use a multi-Frey approach combining two Frey elliptic curves over totally real fields, a Frey hyperelliptic curve over due to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equation for all integers . Moreover, we explain how the use of higher dimensional Frey abelian varieties allows a more efficient proof of this result due to additional structures that they afford, compared to using only Frey elliptic curves. As a second application, we use some of these additional structures that Frey abelian varieties possess to show that a full resolution of the generalized Fermat equation depends only on the Cartan case of Darmon's big image conjecture. In the process, we solve the previous equation for solutions such that and satisfy certain or -adic conditions and all .
Paper Structure (19 sections, 12 theorems, 67 equations, 4 tables)

This paper contains 19 sections, 12 theorems, 67 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Notation and remarks
  3. A Frey curve over $\mathbb{Q}$
  4. Basic properties
  5. Consequences on the solutions
  6. A Frey curve over a totally real cubic field
  7. A first proof of Theorem \ref{['T:main']} using twists of Frey curves
  8. Two multi-Frey approaches to Theorem \ref{['T:main']} using Frey abelian varieties
  9. Proof of Theorems \ref{['T:overQvariety']} and \ref{['T:overQvariety7adic']}
  10. Summary of the results from xhyper_vol1
  11. Proof of Theorem \ref{['T:overQvariety']}
  12. General strategy
  13. Twisted forms
  14. Symmetry
  15. Eliminating the forms
  16. ...and 4 more sections

Key Result

Theorem 1.2

For all integers $n \geq 2$, there are no non-trivial primitive solutions to

Theorems & Definitions (23)

  • Theorem 1.2
  • Theorem 1.4
  • Corollary 1.6
  • Proposition 3.2
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.6
  • proof
  • Proposition 4.3
  • ...and 13 more