Horizontal norm compatibility of cohomology classes for $\mathrm{GSp}_{6}$

TL;DR

This paper develops new horizontal norm relations for motivic cohomology classes related to GSp(6), enabling the construction of Euler systems for associated Galois representations, overcoming previous technical limitations.

Contribution

It introduces a novel approach to establish horizontal norm relations for GSp(6) cohomology classes without relying on multiplicity one, facilitating new Euler systems.

Findings

Established horizontal norm relations for GSp(6) motivic classes.

Applied the relations to construct Euler systems for Galois representations.

Circumvented the failure of multiplicity one hypothesis in the proof.

Abstract

We establish abstract horizontal norm relations involving the unramified Hecke-Frobenius polynomials that correspond under the Satake isomorhpism to the degree eight spinor $L$-factors of $ \mathrm{GSp}_{6} $. These relations apply to classes in the degree seven motivic cohomology of the Siegel modular sixfold obtained via Gysin pushforwards of Beilinson's Eisenstein symbol pulled back on one copy in a triple product of modular curves. The proof is based on a novel approach that circumvents the failure of the so-called multiplicity one hypothesis in our setting, which precludes the applicability of an existing technique. In a sequel, we combine our result with the previously established vertical norm relations for these classes to obtain new Euler systems for the eight dimensional Galois representations associated with certain non-endoscopic cohomological cuspidal automorphic representations of $ \mathrm{GSp}_{6} $.