A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves

In this paper we carry out the steps of Darmon’s program for the generalized Fermat equation $$x^n+y^n=z^5\mathrm{.}$$ In particular, we develop the machinery necessary to prove an optimal bound on the exponent $n$ for solutions satisfying certain $2$-adic and $5$-adic conditions which are natural from the point of view of the method. We also reduce the problem of resolving this equation to a `big image conjecture’, completing a line of ideas suggested in his original program. The above equation is an example of a generalized Fermat equation for which the predicted Frey abelian varieties have dimension $>1$ and thus it represents an interesting test case for Darmon’s program.