### Remarks on the Asymptotic Fermat Conjecture

Linfoot Number Theory Seminar

7th October 2020, 11:00 am – 12:00 pm

Virtual Seminar, https://bristol-ac-uk-dev.zoom.us/j/94649459102

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.

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