The ring of U-operators: Definitions and Integrality

TL;DR

This paper introduces the ring of U-operators for reductive p-adic groups, demonstrating their integrality over the spherical Hecke algebra, which is key for constructing Euler systems and generalizing the Eichler-Shimura relation.

Contribution

It defines and studies the arithmetic of U-operators, showing their integrality over the spherical Hecke algebra, with applications to Euler systems and Shimura varieties.

Findings

U-operators are integral over the spherical Hecke algebra

The operators generalize successor operators for trees

Applications to Euler systems and Eichler-Shimura relation

Abstract

In this paper, we define and study the arithmetic of the ring of $\mathbb{U}$-operators for reductive $p$-adic groups. These operators generalise the notion of "successor" operators for trees with a marked end. We show that they are integral over the spherical Hecke algebra. This integrality intervenes crucially in the construction of Euler systems obtained from special cycles of general Shimura varieties and the generalization of the famous Eichler-Shimura relation.