SOME EXTENSIONS OF THE MODULAR METHOD AND FERMAT EQUATIONS OF SIGNATURE (13, 13, n)

We provide several extensions of the modular method which were moti-vated by the problem of completing previous work to prove that, for any integer n > 2, the equationx13 + y13 = 3znhas no non-trivial primitive solutions. In particular, we present four elimination tech-niques which are based on: (1) establishing reducibility of certain residual Galois representations over a totally real field; (2) generalizing image of inertia arguments to the setting of abelian surfaces; (3) establishing congruences of Hilbert modular forms without the use of often impractical Sturm bounds; and (4) a unit sieve argu-ment which combines information from classical descent and the modular method.The extensions are of broader applicability and provide further evidence that it is possible to obtain a complete resolution of a family of generalized Fermat equations by remaining within the framework of the modular method. As a further illustration of this, we complete a theorem of Anni-Siksek to show that, for P, m > 5, the only primitive solutions to the equation x2` + y2m = z13 are trivial.