# Topological Quantum Field Theory Seminar

### 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 ${x}^{13}+{y}^{13}=C{z}^{p}$ have no solutions $\left(a,b,c\right)$ such that $\mathrm{gcd}\left(a,b\right)=1$ and $13\nmid c$ if $p>4992539$. We will first relate a putative solution of the previous equation to the solution of another Diophantine equation with coefficients in $Q\left(\sqrt{13}\right)$. Then we attach Frey curves $E$ over $Q\left(\sqrt{13}\right)$ to solutions of the latter equation. Finally, we will discuss on the modularity of $E$ 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.
Duration 90 minutes or slightly less