Towards a p-adic nowhere density conjecture of Hecke Orbits

TL;DR

This paper proposes a conjecture about the p-adic nowhere density of Hecke orbits in Shimura varieties and proves that points with large monodromy form an open dense locus in certain neighborhoods.

Contribution

It introduces a new conjecture on p-adic density of Hecke orbits and proves a density result for points with large monodromy in formal neighborhoods.

Findings

Locus with large monodromy is open dense in formal neighborhoods.

Proposes a p-adic nowhere density conjecture for Hecke orbits.

Establishes density results in the context of Shimura varieties.

Abstract

We propose a conjecture on the $p$-adic nowhere density of the Hecke orbit of subvarieties of Hodge type Shimura varieties. By investigating the monodromy of $p$-adic Galois representations associated with points on such Shimura varieties, we prove that the locus in a formal $\mathcal{O}_K$-neighborhood of a mod $p$ point that has large monodromy is open dense, where $K$ is a totally ramified finite extension of $\breve{\mathbb{Q}}_p.$