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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 ${x}^{13}+{y}^{13}=C{z}^{p}$ have no solutions $(a,b,c)$ such that $\mathrm{gcd}(a,b)=1$ 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 $Q(\sqrt{13})$. Then we
attach Frey curves $E$ over $Q(\sqrt{13})$ 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