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The first step to apply the modular method to the equation above is to attach a Frey curve to a putative solution of it. It is known that if we can find multiple Frey curves then maybe we can obtain sharper bounds on the exponent $p$ for which the equation has no solutions satisfying $\|xyz\| > 1$.
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In this talk I will report on ongoing work with Nicolas Billerey, Imin Chen and Luis Dieulefait. We focus on Fermat equations of the form $x^r + y^r = dz^p$, where r is a fixed prime, via the multi-Frey technique. The first step to apply the modular method to the equation above is to attach a Frey curve to a putative solution of it. It is known that if we can find multiple Frey curves then maybe we can obtain sharper bounds on the exponent $p$ for which the equation has no solutions satisfying $|xyz| > 1$. We will discuss other advantages of having multiple Frey curves. Namely, how we can "transfer" irreducibility of the mod $p$ Galois representations from one Frey curve to another and how we can reduce the amount of computations of modular forms required (which sometimes are too large to be feasible). We will sketch how these ideas can be used to attack the equation above in the cases $r=5,13$ and $d=3$.
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