Solving aX^p + bY^p = cZ^p with abc containing an arbitrary number of prime factors
Institut de Mathématiques de Marseille (UMR 7373)
Site Sud - Bâtiment TPR2
Campus de Luminy, Case 907
13288 MARSEILLE Cedex 9
Eduardo SOTO (Universitat de Barcelona)
Let a,b,c be non-zero integers. The Asymptotic Fermat Conjecture (AFC) with coefficients a,b,c predicts that the total set of rational points of aX^p + b Y^p + c Z^p=0 for p ranging over the set of primes can be achieved in a finite set P_a,b,c of exponents p.
This conjecture is connected to the theory of elliptic curves due to Mazur and Frey.
The first cases of the conjecture were stablished by Wiles, Serre, Mazur, Frey, Ribet or Kraus via the modularity method and S-unit equations.
In this talk we will give an introduction to the topic, i.e. elliptic curves, S-unit equations and modularity.
We shall exhibit a further study of the S-unit equation that allows us to prove AFC for abc containing an arbitrary number of prime factors.
This is joint work with Luis Dieulefait (Universitat de Barcelona).