A universal Euler system for GSp(4)

TL;DR

This paper refines the construction of Euler systems for GSp(4) Galois representations, showing they are multiples of a universal class independent of local test data choices.

Contribution

It demonstrates that all Euler system classes for GSp(4) are scalar multiples of a single universal class, clarifying their dependence on local test data.

Findings

Euler system classes lie in a 1-dimensional space

All classes are explicit multiples of a universal class

Dependence on local test data is fully characterized

Abstract

In our earlier work with Christopher Skinner (J. Eur. Math. Soc 24 (2022), no. 2; DOI 10.4171/JEMS/1124; Arxiv 1706.00201), we constructed Euler systems for the 4-dimensional spin Galois representations corresponding to automorphic forms for GSp(4). This construction depended on various arbitrary choices of local test data. In this paper, we use multiplicity-one results for smooth representations to determine how these Euler system classes depend on the choice of test data, showing that all of these classes lie in a 1-dimensional space and are explicit multiples (given by local zeta-integrals) of a "universal" class independent of the choice of test data.