On an approach to automorphic Euler systems

Abstract

We describe a novel approach towards establishing Hecke algebra valued horizotal norm relations between push-forwards of integral cohomology classes that appear in motives associated with Shimura varieties. These relations then give rise to Euler systems for appropriate automorphic Galois representations that arise in the cohomology of such varieties. The key innovation in our approach is a precise axiomatic formulation of a refined relation that yields the desired horizontal norm relations in a universal sense. A crucial ingredient in the execution of this approach is a systematic and explicit decomposition recipe for Hecke operators of unramified reductive groups over local fields obtained using Bruhat-Tits theory. We use our approach to establish such relations in a variety of examples, some previously studied and some new.