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The modular approach to Diophantine equations over totally real fields

Maleeha Khawaja, Samir Siksek

TL;DR

The paper surveys the extension of Wiles’ modular approach to Diophantine equations from $\mathbb{Q}$ to totally real fields, detailing obstacles such as partial level-lowering results and the need for Hilbert modular forms. It presents the Frey-curve framework, Galois representations, and modularity lifting/switching as core tools, and outlines techniques like multi-Frey and Frey hyperelliptic curves to attack generalized Fermat equations. A central theme is the systematic elimination of newforms (newform elimination) to obtain exponent bounds and contradictions, enabling Fermat-type results over real quadratic, cubic, quartic, and higher-degree totally real fields. The article also highlights key advances in modularity over totally real fields, including finite-nonmodular-j-invariant results, and discusses how these developments yield new avenues for resolving Fermat-type equations in broader arithmetic settings. Overall, it positions modularity and level-lowering as a cohesive framework for understanding Diophantine equations over totally real fields and outlines significant progress across degrees up to quartic and beyond.

Abstract

Wiles' proof of Fermat's last theorem initiated a powerful new approach towards the resolution of certain Diophantine equations over $\mathbb{Q}$. Numerous novel obstacles arise when extending this approach to the resolution of Diophantine equations over totally real number fields. We give an extensive overview of these obstacles as well as providing a survey of existing methods and results in this area.

The modular approach to Diophantine equations over totally real fields

TL;DR

The paper surveys the extension of Wiles’ modular approach to Diophantine equations from to totally real fields, detailing obstacles such as partial level-lowering results and the need for Hilbert modular forms. It presents the Frey-curve framework, Galois representations, and modularity lifting/switching as core tools, and outlines techniques like multi-Frey and Frey hyperelliptic curves to attack generalized Fermat equations. A central theme is the systematic elimination of newforms (newform elimination) to obtain exponent bounds and contradictions, enabling Fermat-type results over real quadratic, cubic, quartic, and higher-degree totally real fields. The article also highlights key advances in modularity over totally real fields, including finite-nonmodular-j-invariant results, and discusses how these developments yield new avenues for resolving Fermat-type equations in broader arithmetic settings. Overall, it positions modularity and level-lowering as a cohesive framework for understanding Diophantine equations over totally real fields and outlines significant progress across degrees up to quartic and beyond.

Abstract

Wiles' proof of Fermat's last theorem initiated a powerful new approach towards the resolution of certain Diophantine equations over . Numerous novel obstacles arise when extending this approach to the resolution of Diophantine equations over totally real number fields. We give an extensive overview of these obstacles as well as providing a survey of existing methods and results in this area.
Paper Structure (36 sections, 33 theorems, 125 equations, 2 tables)

This paper contains 36 sections, 33 theorems, 125 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Level lowering and the Frey curve $E$
  3. Level lowering
  4. The Frey curve
  5. The $(p,p,p)$ Frey curve
  6. The $(p,p,2)$ and $(p,p,3)$ Frey curves
  7. The multi-Frey approach
  8. Frey hyperelliptic curves
  9. Galois representations associated to elliptic curves
  10. Definition and first examples
  11. An Example: $\overline{\rho}_{E,2}$
  12. The mod $N$-cyclotomic character
  13. Torsion and isogenies
  14. Quadratic twisting
  15. Local properties of mod $N$ representations of elliptic curves
  16. ...and 21 more sections

Key Result

Theorem 1

Let $n\geq 3$ be an integer. If $(a,b,c)\in \mathbb{Q}^3$ is a solution to the Fermat equation then $abc=0$.

Theorems & Definitions (57)

  • Theorem 1: Wiles
  • Theorem 2: Wiles, Breuil, Conrad, Diamond and Taylor
  • Theorem 3: Caraiani and Newton
  • Theorem 4: Ribet
  • Theorem 5: Jarvis, Fujiwara and Rajaei
  • Example 8
  • Example 9
  • Theorem 10: Billerey, Chen, Dieulefait and Freitas
  • Theorem 11: Billerey, Chen, Dieulefait and Freitas
  • Theorem 12
  • ...and 47 more