I have been told by some people that localglobal Langlands compatibility for $GL_2$ (the vanilla version, not the one being developed in this decade by Emerton and others) implies Shimura conjecture on reduction of the quotient of Jacobians of modular curves. How does this implication work? There are also some stronger results of KatzMazur, I think. Can Langlands be applied to prove them?
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$\begingroup$ I believe the following notes by Daniel Litt address your main question: daniellitt.com/s/localglobal.pdf $\endgroup$ – Jesse Silliman May 13 at 2:16

$\begingroup$ @JesseSilliman I am not sure they do so completely. The notes don't cite KatzMazur. Maybe the answer is easy to infer from that for someone experienced, not for me though. $\endgroup$ – schematic_boi May 13 at 11:14

$\begingroup$ What statements in KatzMazur are you interested in? It would be easier if you asked for the proof of a specific statement. $\endgroup$ – Will Sawin May 13 at 13:57

$\begingroup$ @WillSawin I am trying to make sense of a comment (mathoverflow.net/questions/171831/…), I am not quite sure about the exact theorem yet. Will try to figure it out. $\endgroup$ – schematic_boi May 13 at 14:17

$\begingroup$ I admit I was not sure what theorem you were interested in, but those notes describe how to use localglobal compatibility to prove that the quotient $J_1(p)/J_0(p)$ has good reduction over $\mathbb{Q}(\zeta_p)^+$. I don't know if this is also what KatzMazur proves in its final chapter. These notes point out that it is not clear if the localglobal proof is circular, as you need geometric input to prove localglobal compatibility. I think Coleman gave another proof of this result by constructing a semistable model for $X_1(p)$ over this field. $\endgroup$ – Jesse Silliman May 13 at 16:10