Seed Relations for Eichler--Shimura congruences and Euler systems

TL;DR

This paper establishes a key algebraic property of the $ ext{U}$-operator related to cocharacters, which is crucial for advancing the understanding of norm relations and Eichler--Shimura relations in number theory.

Contribution

It proves that the $ ext{U}$-operator is a right root of the Hecke polynomial, enabling new developments in Euler systems and congruences related to Eichler--Shimura theory.

Findings

$ ext{U}$-operator is a right root of the Hecke polynomial

Supports proof of horizontal norm relations for Gross--Gan--Prasad cycles

Facilitates generalization of Eichler--Shimura relations

Abstract

This paper proves that the $\mathbb{U}$-operator attached to a cocharacter is a right root of the corresponding Hecke polynomial. This result is an important ingredient in the proof of (i) the horizontal norm relations in the context of Gross--Gan--Prasad cycles and of (ii) the generalization of Eichler--Shimura relations.