# Finite length results for the mod p cohomology of GL_2

# Finite length results for the mod p cohomology of GL_2

Suppose that F/Q_p is a finite unramified extension. When F = Q_p, Emerton showed that the mod-p local Langlands correspondence is realized globally in the mod-p cohomology of modular curves. For larger F, the mod-p local Langlands correspondence of GL_2(F) is unknown, but it is interesting to study the subrepresentations of the mod-p cohomology of Shimura curves cut out by a global Galois representation, in analogy with F = Q_p. Under some reasonable hypotheses we prove that these representations of GL_2(F) are of finite length, as was expected since the work of Breuil-Paskunas from around 2007. This is joint work with C. Breuil, Y. Hu, S. Morra, and B. Schraen.

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