Abstract: Let p be a prime number, K a finite unramified extension of Qp, and pi a smooth representation of GL2(K) on some Hecke eigenspace in the H^1 mod p of a Shimura curve. One can associate to pi a multivariable (phi, O_K*)-module D_A(pi). I will state a conjecture which describes D_A(pi) in terms of the underlying 2-dimensional mod p representation of Gal(Kbar/K). When the latter is semi-simple (sufficiently generic), I will sketch a proof of this conjecture. This is joint work with F. Herzig, Y. Hu, S. Morra and B. Schraen.
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