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be a totally real field in which
is unramified. Let
be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place
be the corresponding Hecke eigensystem. We describe the
-torsion in the
cohomology of Shimura curves with full congruence level at
-representation. In particular, it only depends on
and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic
-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].
We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a
-arithmetic manifold is purely local, that is, only depends on the Galois representation at places above
. This is a generalization to
of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights
as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame
-dimensional Galois representations’, Duke Math. J.149(1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math.212(1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group
is a CM extension of number fields in which the prime
splits completely and every other prime is unramified. Fix a place
. Suppose that
is a continuous irreducible Galois representation such that
is upper-triangular, maximally non-split, and generic. If
is automorphic, and some suitable technical conditions hold, we show that
can be recovered from the
-action on a space of mod
automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for
, show the existence of an ordinary lifting of
, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations
to which our main theorem applies.
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