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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 which satisfies the Taylor–Wiles hypotheses and is generic
at a place
v
above
p .
Let
m
be the corresponding Hecke eigensystem. We show that the
m -torsion
in the
mod
p
cohomology of Shimura curves with full congruence level at
v coincides with the
GL 2 ( k v ) -representation
D 0 ( r ̄ | G F v ) constructed
by Breuil and Paškūnas. In particular, it depends only on the local representation
r ̄ | G F v , and its
Jordan–Hölder factors appear with multiplicity one. This builds on and extends work of
the author with Morra and Schraen and, independently, Hu–Wang, which proved these
results when
r ̄ | G F v
was additionally assumed to be tamely ramified. The main new tool is a method for
computing Taylor–Wiles patched modules of integral projective envelopes using
multitype tamely potentially Barsotti–Tate deformation rings and their intersection
theory.
Keywords
Galois deformations, mod p Langlands program
Mathematical Subject Classification 2010
Primary: 11S37
Milestones
Received: 19 October 2017
Revised: 13 February 2019
Accepted: 27 May 2019
Published: 9 October 2019