Diana Mocanu

The modularity approach over totally real fields

I will use this occasion to 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 \textit{signature} of the equation. We will see various results concerning the solutions to the Fermat equation with signature $(p,p,2)$ and $(p,p,3)$ using image of inertia comparison and the study of $S$-unit equations. This is a completed project under the supervision of Samir Siksek. If time permits, I will show some ongoing work on how one can approach the more difficult family of equations of signatures $(r,r,p)$ (fixed r).