Martin Azon

Effective Darmon's program for the generalised Fermat equation

In 2000, Darmon presented a program to solve infinite families of generalised Fermat equations $Ax^p + By^q = Cz^r$, where some of the exponents vary. In this talk, I will present the ideas of Darmon's program, and I will solve instances of families of GFEs with non-trivial coefficients. First, I will recall the construction of Frey hyperelliptic curves attached to a putative solution, and I will introduce a common framework allowing for a uniform treatment of different families of GFEs. Using the theory of cluster pictures and other geometric techniques, I will describe the reduction types of Jacobians of Frey curves. After this, I will discuss the main properties of the associated Galois representations (modularity, irreducibility, level lowering). The end of the talk will deal with numerical computations to contradict the existence of solutions. Part of this talk is based on joint work with Mar Curcó-Iranzo, Maleeha Khawaja, Céline Maistret and Diana Mocanu.