We use the Taylor–Wiles–Kisin patching method to investigate the
multiplicities with which Galois representations occur in the mod
cohomology of Shimura curves over totally real number fields. Our method relies on
explicit computations of local deformation rings done by Shotton, which we use
to compute the Weil class group of various deformation rings. Exploiting
the natural self-duality of the cohomology groups, we use these class group
computations to precisely determine the structure of a patched module in many
new cases in which the patched module is not free (and so multiplicity one
fails).
Our main result is a “multiplicity
”
theorem in the minimal level case (which we prove under some mild technical hypotheses),
where
is a number that depends only on local Galois theoretic information at the primes
dividing the discriminant of the Shimura curve. Our result generalizes Ribet’s
classical multiplicity 2 result and the results of Cheng, and provides progress towards
the Buzzard–Diamond–Jarvis local-global compatibility conjecture. We also
prove a statement about the endomorphism rings of certain modules over the
Hecke algebra, which may have applications to the integral Eichler basis
problem.
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