On constructing zeta elements for Shimura varieties

TL;DR

This paper introduces a new axiomatic approach for constructing Euler systems from Shimura varieties, enabling applications to more general cases including non-spherical pairs and complex Galois representations.

Contribution

It develops a unified framework for horizontal norm relations in Euler systems applicable to various classes of Shimura varieties and Galois representations.

Findings

Framework applies to both algebraic cycles and Eisenstein classes.

Enables construction of Euler systems for spinor Galois representations.

Extends methods beyond local multiplicity one hypotheses.

Abstract

We present a novel axiomatic framework for establishing horizontal norm relations in Euler systems that are built from pushforwards of classes in the motivic cohomology of Shimura varieties. This framework is uniformly applicable to the Euler systems of both algebraic cycles and Eisenstein classes. It also applies to non-spherical pairs of groups that fail to satisfy a local multiplicity one hypothesis, and thus lie beyond the reach of existing methods. A key application of this work is the construction of an Euler system for the spinor Galois representations arising in the cohomology of Siegel modular varieties of genus three, which is undertaken in two companion articles.