Room P4.35, Mathematics Building
Nuno Freitas, Univ. Barcelona
Fermat-type equations of signature \((13,13,p)\) via Hilbert cuspforms
In this talk I will give an introduction to the modular approach to Fermat-type equations via Hilbert cuspforms and discuss how it can be used to show that certain equations of the form have no solutions such that and if . We will first relate a putative solution of the previous equation to the solution of another Diophantine equation with coefficients in . Then we attach Frey curves over to solutions of the latter equation. Finally, we will discuss on the modularity of and irreducibility of certain Galois representations attached to it. These ingredients enable us to apply a modular approach via Hilbert newforms to get the desired arithmetic result on the equation.