An image of inertia argument for abelian surfaces and Fermat equations of signature (13,13,n).

Building on previous work, we show that, for any integer $n geq 2$, the equation $$x^{13} + y^{13} = 3 z^n$$ has no non-trivial solutions. For this, we need to deal with the obstruction which arises from the fact that the $7$-torsion of one of the Frey curves associated to this equation is a Galois submodule of the $7$-torsion of the Jacobian of a certain genus $2$ hyperelliptic curve~$C$. We remove this obstruction by combining the modularity of the Jacobian of $C$ with an `image of inertia argumentu0027 applied to that surface.