@SamNead The lecture contains an extensive discussion of the difficulty Wiles had and their epistemological implications but, as I said, I wanted to be able to seriously discuss very stringent epistemological criteria and how they apply (such criteria would immediately invalidate partial proofs or ongoing works).

They have the same $L$-function (locally and globally), which is both pretty impressive and quite far from being "basically the same".

numdam.org/article/SB_1968-1969__11__139_0.pdf is the historical source ams.org/journals/bull/1995-32-04/S0273-0979-1995-00616-6/… is a standard reference ms.u-tokyo.ac.jp/~t-saito/talk/eepr.pdf is also good

Galois extensions of local fields are solvable (this follows from the inertia filtration)

@plm No idea, honestly. Maybe Jack Thorne will show up (by the way, I'm not the Olivier you are apparently thinking of).

@user145307 Oh yeah, you are right! Above, I was answering a totally different question, namely whether the Hecke algebra generated by operators outside the level $p$ and some auxiliary prime $q$ is equal to the full Hecke algebra (answer is no, one cannot always recover $T_2$ when $q=2$) whereas the question of the OP was whether one can recover $U_p$, not $T_q$. Thank you for the correction, I'll edit.