Cohomology of quaternionic Shimura varieties at parahoric levels (Q1601089)

Let \(Sh_D\) be the Shimura variety associated to the multiplicative group \(D^\times\) of a quaternion division algebra \(D\) over a totally real number field \(F\). In this paper the author proves an identity between some semisimple \(L\)-functions associated to the cohomology of \(Sh_D\) and semisimple automorphic \(L\)-functions. By assuming a certain purity condition he also obtains a similar result for the usual \(L\)-functions. The main theorem in this paper extends the results in a previous paper by the author [Math. Ann. 317, 41-55 (2000; Zbl 1009.11045)] to a more general case.