## STNB2020(34th edition)

### A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves

Imin Chen

#### Abstract

We will explain how the ideas and methods developed in previous lectures can be continued to study Fermat type equations of signature $(p,p,5)$ in joint work with Angelos Koutsianas. Specifically, we will give a resolution of the equation $x^n + y^n = z^5$ in the congruence classes mod $10$ which avoid the trivial solutions. The approach uses two Frey hyperelliptic curves of genus $2$ introduced by Darmon, ideas from his program, and extensions of the methods described in the previous two lectures.