In the Bennet, Chen, Dahmen, Yazdani paper, Generalized Fermat equations: A miscellany, on page 24 in section 6 entitled "Future Work", they say:

"A limitation of the modular method at present is that the possible exponents $(p, q, r)$ must relate to a moduli space of elliptic curves (or more generally, abelian varieties of $GL_2$-type). For general $(p, q, r)$, this is not the case."

Is there an example of a $(p,q,r)$ case where Darmon's program must fail? General $(p,q,r)$ is the whole ballgame. If all the conjectures that Darmon uses were proven, then what $(p,q,r)$ case remains that they might be referring to? What $(p,q,r)$ use case do they have in mind?



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