Seed Relations for Eichler--Shimura congruences and Euler systems

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.