Multiplicity one for the mod p cohomology of Shimura curves: the tame case

Abstract

Let F be a totally real field, p an unramified place of F dividing p and r : Gal(F/F) → GL2(Fp) a continuous irreducible modular representation. The work of Buzzard, Diamond and Jarvis (Duke Math Journal, 2010) associates to r an admissible smooth representation of GL2(Fp) on the mod p étale cohomology of Shimura curves attached to indefinite division algebras which split at p. When r satisfies the Taylor–Wiles hypotheses and r|Gal(Fp/Fp) is tamely ramified and generic, we determine the subspace of invariants of this representation under the principal congruence subgroup of level p. As a consequence, this subspace depends only on r|Gal(Fp/Fp) and satisfies a multiplicity one property.