Let be a totally real field in which is unramified. Let be a modular Galois representation which satisfies the Taylor–Wiles hypotheses and is generic at a place above . Let be the corresponding Hecke eigensystem. We show that the -torsion in the cohomology of Shimura curves with full congruence level at coincides with the -representation constructed by Breuil and Paškūnas. In particular, it depends only on the local representation , and its Jordan–Hölder factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and, independently, Hu–Wang, which proved these results when was additionally assumed to be tamely ramified. The main new tool is a method for computing Taylor–Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti–Tate deformation rings and their intersection theory.
"Multiplicity one for wildly ramified representations." Algebra Number Theory 13 (8) 1807 - 1827, 2019. https://doi.org/10.2140/ant.2019.13.1807