Correspondences on Shimura curves and Mazur’s principle at p

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The Fermat equation over Q(2) (2004, methodology): “We may now remove the primes above p using the main result of [ 20 ] (working still with Shimura curves on this quaternion algebra), and finally remove the auxiliary prime using Fujiwara’s version of Mazur’s Principle for even degree [11].”
On Serre's conjecture for 2-dimensional mod p representations of Gal( Q=Q) (2009, methodology): “We have proved the proposition if k(¯ ρ) = 2. If we proceed this way, we may obviate the use of level-lowering results o f [28] at the end of the proof of the proposition.)”