Let F be a totally real field, v an unramified place of F dividing p and rhoa continuous irreducible two-dimensional mod p representation of G_F such thatthe restriction of rho to G_{F_v} is reducible and sufficiently generic. If rhois modular (and satisfies some weak technical assumptions), we show how torecover the corresponding extension between the two characters of G_{F_v} interms of the action of GL_2(F_v) on the cohomology mod p.
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