We use the Taylor–Wiles–Kisin patching method to investigate the multiplicities with which Galois representations occur in the mod $\ell$ 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 $2^k$” theorem in the minimal level case (which we prove under some mild technical hypotheses), where $k$ 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.

我们使用Taylor–Wiles–Kisin修补方法来研究mod中Galois表示出现的多重性 $\ell$Shimura曲线在完全实数域上的同调性。我们的方法依赖于Shotton完成的局部变形环的显式计算，我们将其用于计算各种变形环的Weil类组。利用同调组的自然对偶性，我们使用这些类组计算来精确确定修补模块在很多情况下的结构，这些新情况中修补模块不是自由的（因此多重性会失败）。

我们的主要结果是“多重性” ${\mathrm{2个}}^{ķ}$最小定理（在一些温和的技术假设下证明）的定理，其中 $ķ$是仅取决于局部Shilo曲线判别式上的局部Galois理论信息的数字。我们的结果推广了Ribet的经典多重性2结果和Cheng的结果，并为Buzzard–Diamond–Jarvis局部-全局兼容性猜想提供了进展。我们还证明了有关Hecke代数上某些模块的内同态环的陈述，这可能适用于积分Eichler基问题。