Effective Darmon's program for the generalised Fermat equation

Abstract

We follow the ideas of Darmon's program for solving infinite families of generalised Fermat equations of signatures $(p,p,r)$ and $(r,r,p)$, where, $r$ is a fixed prime and $p$ is varying. We do so by introducing a common framework for both signatures, allowing for a uniform treatment for the two families of equations. We analyse in detail the geometry of Frey hyperelliptic curves, and the reduction types of the N\'eron models of their Jacobians. We then study the associated $2$-dimensional Galois representations: modularity, irreducibility, and level lowering. We provide a Magma package that performs the elimination step for many choices of coefficients and the exponent $r$. In order to illustrate the effectiveness of our results, we solve several examples of families of equations of signatures $(p,p,5)$ and $(5, 5, p)$.