STNB2020(34th edition)

A multi-Frey approach to Fermat equations of signature~$(7,7,p)$ combining Frey varieties of different dimensions.

Nuno Freitas

Abstract

We will discuss how various ideas from previous lectures combine to study Fermat type equations of signature $(7,7,p)$. More precisely, using a multi-Frey approach with two Frey elliptic curves over totally real fields, a Frey hyperelliptic over~$\Q$ due to Kraus and ideas from the Darmon program, we will give a complete resolution of the equation $$x^7 + y^7 = 3 z^n$$ for all integers $n \ge 2$.