STNB2020(34th edition)

Galois Representations and Diophantine Equations


Luis Dieulefait and Nuno Freitas


One of the greatest mathematical breakthroughs of the 20th century is Wiles' proof of modularity of semistable elliptic curves over Q, which completed the proof of Fermat's Last Theorem. This ushered what is now called the modular method to Diophantine equations. The idea, originally due to Frey, Serre, Ribet and Wiles is to attach to a putative solution of a Diophantine equation an elliptic curve E (known as a Frey curve), and study the mod p Galois representation attached to E via modularity and level lowering. This relates the solution to a modular form of weight 2 and small level and, to conclude the putative solution does not exist, one needs to show that such relation leads to a contradiction. In this series of lectures we will discuss the modular method and some of its recent generalizations such as extensions to number fields and Darmon's program to attack the Generalized Fermat Equation which replaces the classical Frey curve by higher dimension abelian varieties.


  1. Introduction to the modular method (Nicolas Billerey)
  2. On the Asymptotic Fermat Conjecture (Samir Siksek)
  3. Fermat's Last Theorem and Wendt's Resultant (Alain Kraus)
  4. The Darmon program for the Generalized Fermat equation (Luis Dieulefait)
  5. A multi-Frey approach to Fermat equations of signature~$(7,7,p)$ combining Frey varieties of different dimensions. (Nuno Freitas)
  6. A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves (Imin Chen)