The plectic conjecture over local fields

TL;DR

This paper proves a local version of the plectic conjecture for Shimura varieties using a mixed-characteristic fusion approach, and establishes a $p$-adic uniformization theorem for certain Shimura varieties.

Contribution

It introduces a mixed-characteristic fusion method to prove the plectic conjecture locally and extends results to global Shimura varieties via decomposition groups.

Findings

Proved an analog of the plectic conjecture for local Shimura varieties.

Established a $p$-adic uniformization theorem for the basic locus of abelian type Shimura varieties.

Connected local and global plectic conjectures through new uniformization techniques.

Abstract

Using a mixed-characteristic incarnation of fusion, we prove an analog of Nekov\'a\v{r}-Scholl's plectic conjecture for local Shimura varieties. We apply this to obtain results on the plectic conjecture for (global) Shimura varieties after restricting to a decomposition group. Along the way, we prove a $p$-adic uniformization theorem for the basic locus of abelian type Shimura varieties at hyperspecial level, which is of independent interest.