On the geometric connected components of moduli spaces of p-adic shtukas and local Shimura varieties

Abstract

We study topological properties of moduli spaces of p-adic shtukas and local Shimura varieties. On one hand, we construct and study the specialization map for moduli spaces of p-adic shtukas at parahoric level whose target is an affine Deligne-Lusztig variety. On the other hand, given a p-adic shtuka datum $(G,b,\mu )$, with $G$ unramified over ${\mathbb{Q}}_p$ and such that $(b,\mu )$ is HN-irreducible, we determine the set of geometric connected components of infinite level moduli spaces of p-adic shtukas. In other words, we understand ${\pi }_0({\mathrm{Sht}}_{(G,b,\mu ,\infty )}\times \mathrm{Spd}{\mathbb{C}}_p)$ with its right $G({\mathbb{Q}}_p)\times G_b(\mathbb{Q}p)\times W_E$ -action. As a corollary, we prove new cases of a conjecture of Rapoport and Viehmann.

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