On Darmon's program for the generalized Fermat equation, I

TL;DR

This work advances Darmon's program for the generalized Fermat equation by wiring together Frey representations of signatures $(p,p,r)$ and $(r,r,p)$ through Kraus’ hyperelliptic curves, proving GL$_2$-type modularity for the associated Jacobians over the totally real field $K=\mathbb{Q}(\zeta_r)^+$. It develops a comprehensive toolkit—conductor calculations, irreducibility criteria, finiteness criteria for residual representations, and refined level-lowering with prescribed inertial types—to perform elimination against Hilbert newforms, culminating in Cartan-case reductions and reductions to CM forms in favorable cases. These advances yield concrete Diophantine applications, notably proving the nonexistence of nontrivial primitive solutions to $x^{11}+y^{11}=z^n$ for all $n\ge2$ under $2\mid a+b$ or $11\mid a+b$, and enabling a multi-Frey strategy that improves computational tractability for large coefficient fields. The combination of hyperelliptic Frey curves with modern modularity lifting over totally real fields broadens the reach of the modular method and opens pathways to resolve broader Fermat-type equations and tighten elimination of non-CM Hilbert newforms.

Abstract

In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of $\mathrm{GL}_2$-type over totally real fields. The original approach was based on hard open conjectures, which have made it difficult to apply in practice. In this paper, building on the progress surrounding the modular method from the last two decades, we analyze and expand the current limits of this program by developing all the necessary ingredients to use Frey abelian varieties for new Diophantine applications. In particular, we deal with all but the fifth and last step in the modular method for Fermat equations of signature $(r,r,p)$ in almost full generality. As an application, for all integers $n \geq 2$, we give a resolution of the generalized Fermat equation $x^{11} + y^{11} = z^n$ for solutions $(a,b,c)$ such that $a + b$ satisfies certain $2$- or $11$-adic conditions. Moreover, the tools developed can be viewed as an advance in addressing a difficulty not treated in Darmon's original program: even assuming `big image' conjectures about residual Galois representations, one still needs to find a method to eliminate Hilbert newforms at the Serre level which do not have complex multiplication. In fact, we are able to reduce the problem of solving $x^5 + y^5 = z^p$ to Darmon's `big image conjecture', thus completing a line of ideas suggested in his original program, and notably only needing the Cartan case of his conjecture.

Theorems & Definitions (135)

• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.5
• Remark 1.7
• Theorem 1.8
• Corollary 1.9
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof