$\mathcal{G}$-$\mu$-displays and local shtuka

TL;DR

This paper compares two methods for constructing integral models of local Shimura-varieties, demonstrating their equivalence and proving key representability conjectures using advanced algebraic structures.

Contribution

It establishes the equivalence between $ ext{G}$-$ ext{mu}$-displays and local shtuka approaches and proves the local representability conjecture of Rapoport-Pappas.

Findings

Proved the representability of the diamond of the generic fiber.

Verified the local representability conjecture of Rapoport-Pappas.

Applied methods to all unramified local Shimura-data with certain slope conditions.

Abstract

Two approaches to the construction of integral models of local Shimura-varieties are compared: that of B\"ultel-Pappas using $\mathcal{G}$-$\mu$-displays and that of Scholze using local mixed-characteristic shtuka. As an application, the representability of the diamond of the generic fiber of the B\"ultel-Pappas moduli problem is verified and a local representability conjecture of Rapoport-Pappas is proven. The methods apply to all unramified local Shimura-data under a certain condition on slopes and if $p\neq 2;$ thus in particular also to exceptional groups.