We study motivic amplitudes--objects which contain all of the
Essential mathematical content of scattering amplitudes in planar SYM theory in
A completely canonical way, free from the ambiguities inherent in any attempt
To choose particular functional representatives. We find that the cluster
We prove conjectures of Breuil and Breuil-Dembele (C. Breuil, "Sur un
Probleme de compatibilite local-global modulo p pour GL(2)"), including a
Generalisation from the principal series to the cuspidal case, subject to a
Mild global hypothesis that we make in order to apply certain R=T theorems.
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