The multiplicity of Hecke eigenspaces in the mod p cohomology of Shimura curves is a classical invariant which has been computed in significant generality when the group splits at p. These results have recently found interesting applications to the mod p Langlands correspondence for GL_2 over unramified p-adic fields. As a first step towards extending these to nonsplit quaternion algebras, we prove a new multiplicity one theorem in the nonsplit case. The main idea of the proof is to use the Breuil--Kisin module associated to a finite flat model of the cohomology to reduce the problem to a known statement about modular forms on totally definite quaternion algebras. Speaker(s): Andrea Dotto (University of Chicago)
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