S=T for Shimura Varieties and Moduli Spaces of p-adic Shtukas

TL;DR

This paper proves the $S=T$ conjecture for Shimura varieties and moduli spaces of p-adic Shtukas using advanced geometric and categorical tools, and derives the Eichler--Shimura relation for certain Shimura varieties.

Contribution

It establishes the $S=T$ conjecture for Shimura varieties using Scholze's theory and Fargues--Scholze's geometric Satake, advancing the understanding of their arithmetic geometry.

Findings

Proof of the $S=T$ conjecture for Shimura varieties

Derivation of the Eichler--Shimura relation for Hodge type Shimura varieties

Application of diamonds, v-stacks, and geometric Satake in this context

Abstract

We prove the $S=T$ conjecture proposed by Xiao--Zhu in \cite{2017arXiv170705700X}, making use of Scholze's theory of diamonds and v-stacks and Fargues--Scholze's geometric Satake equivalence. Following \cite{2018arXiv180205299X}, we deduce the Eichler--Shimura relation for Shimura varieties of Hodge type.