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 simple and simpy connected. The proof uses a Hecke algebra associated with $G(K((t)))$ and a global argument involving $G$-bundles on curves.

34 pages; v2: Proposition 4.8 corrected; v3: 37 pages; auxiliary results on G-bundles over arbitrary fields are strengthened and moved to the appendix written by the third and the fourth authors; v5: 40 pages; corrected a mistake in the proof of Lemma 5.8 (formerly Lemma 5.7)