Imagen: Álvaro Minguito/ICMAT

For more than three centuries, Fermat’s Last Theorem was one of the most famous and elusive problems in mathematics. Formulated in the seventeenth century, it states that the equation x^n+y^n=z^n has no non-trivial integer solutions x,y,z when n>2. Its definitive proof came in 1994-95, when Andrew Wiles solved it by combining deep ideas from number theory, algebraic geometry, and complex analysis. Beyond settling a historic problem, Wiles’s work inaugurated a powerful new strategy for studying so-called Diophantine equations—polynomial equations whose solutions are sought in integers—known as the modular method.

The central idea of this method is to translate a putative solution of a Diophantine equation into an apparently very different object: a Galois representation. These representations describe deep algebraic symmetries and encode how the solutions of polynomial equations are related to one another. In the context of the modular method, each possible solution is associated with a two-dimensional Galois representation with values in 2×2 matrices whose entries are the integers modulo a prime number. The goal is to show that these representations must arise from very well-classified analytic objects, namely modular forms, which in many cases leads to a contradiction and allows one to rule out the existence of solutions.

In the proof of Fermat’s Last Theorem, the bridge between Diophantine equations and Galois representations was provided by the so-called Frey elliptic curves. However, when one moves to more general equations—such as the generalized Fermat equations of the form x^r+y^q=z^p—this strategy is no longer sufficient. In 2000, the mathematician Henri Darmon proposed an ambitious program to extend the modular method by replacing elliptic curves with more complex objects: abelian varieties of GL2-type defined over totally real fields. Although conceptually very promising, this approach relied on deep conjectures that have made its practical application difficult for many years.

A new work now takes a decisive step in this direction. In it, Nicolas Billerey (Université Clermont Auvergne), Imin Chen (Simon Fraser University), Luis Dieulefait (University of Barcelona), and Nuno Freitas (ICMAT-CSIC) develop and refine the tools needed to effectively apply the modular method based on Frey abelian varieties to new Diophantine problems. In particular, they are able to treat almost completely the Fermat equations x^r+y^r=z^p, significantly expanding the scope of Darmon’s program.

Moreover, the work addresses an important technical difficulty that had not been resolved in the original framework: even under favorable hypotheses on residual Galois representations, one still needs to eliminate certain modular forms—namely Hilbert modular forms without complex multiplication—that do not fit into the classical modular method scheme. The new techniques developed begin to offer solutions to this problem.

As a concrete application of these advances, the authors obtain new results for the equations x^r+y^r=z^p, in the cases r=5 and r=11, providing new proofs of the non-existence of non-trivial integer solutions over wide ranges of exponents.

Reference

Nicolas Billerey, Imin Chen, Luis Dieulefait, Nuno Freitas. On Darmon’s program for the generalized Fermat equation, I. Journal für die reine und angewandte Mathematik, In press, ⟨10.1515/crelle-2025-0014⟩. ⟨hal-04421240⟩