Évènement

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 $K$.