Hecke Operators for Curves Over Non-Archimedean Local Fields and Related Finite Rings

Abstract We study Hecke operators associated with curves over a non-archimedean local field $K$ and over the rings $O/\mathfrak{m}^{N}$, where $O\subset K$ is the ring of integers. Our main result is commutativity of a certain “small” local Hecke algebra over $O/\mathfrak{m}^{N}$, associated with a connected split reductive group $G$ such that $[G,G]$ is simply connected. The proof uses a Hecke algebra associated with $G(K(\!(t)\!))$ and a global argument involving $G$-bundles on curves.