The modular approach for solving $x^r+y^r=z^p$ over totally real number fields

will first introduce the modular method for solving Diophantine Equations, famously used to prove the Fermat Last Theorem. Then, we will see how to generalize it for a totally real number field $K$ and a Fermat-type equation $Aa^p+Bb^q=Cc^r$ over $K$. We call the triple of exponents $(p,q,r)$ the signature of the equation. We will see various results concerning the solutions to the Fermat equation with signatures $(r,r,p)$ (fixed r). This will involve image of inertia comparison and the study of certain $S$-unit equations over $K$.