Edu Soto

Remarks on the Asymptotic Fermat Conjecture

Let a, b, c non-zero integers and consider the family of projective plane curves
C_p: a x^p + b y^p + c z^p = 0, for every prime p. The Asymptotic Fermat Conjecture (AFC)
predicts that there is a constant k = k(a,b,c) such that C_p has no rational point other
than x=\pm y = \pm z for every p> k. In this talk we introduce the problem and describe
the modular method (due to Frey-Mazur and Kraus) typically used to solve some instances
of the conjecture. The method connects the AFC with some S-unit equations, where
S = rad(2abc). We will establish new instances of the conjecture by solving some particular
families of S-unit equations.