Multiplicities of Galois Representations in Cohomology Groups of Shimura Curves

We prove a multiplicity free result in this paper. Specifically, we show that the first cohomology group of a Shimura curve is finite free over the Hecke algebra after localization at a non-Eisenstein ideal. The main tool we use is Diamond's refined Taylor-Wiles system. We also compute the rank of this free module in some cases. In particular, we show that the rank is two under certain technical conditions.