MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL. (15th March 2022) - Le, Daniel; Morra, Stefano; Schraen, Benjamin
Abstract: Let F be a totally real field in which p is unramified. Let r:G_F→ GL_2(F_p) be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place v above p . Let m be the corresponding Hecke eigensystem. We describe the m -torsion in the mod p cohomology of Shimura curves with full congruence level at v as a GL_2(k_v) -representation. In particular, it only depends on r|_I_F_v and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic GL_2(F_q) -projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math. 200 (1) (2015), 1–96].