Moduli stacks of generalized phi-modules

Abstract

Let $F$ be an arbitrary $p$-adic field and let $G$ be an arbitrary reductive group over $F$ with Langlands dual group ${}^{L}G$. We show that the change-of-group morphism of Emerton-Gee stacks ${\mathcal{X}}_{{}^{L}G}\to {\mathcal{X}}_{G{L}_d}$ is relatively representable by algebraic stacks of finite presentation over $\mathrm{Spf}{\mathbf{Z}}_p$ for any embedding ${}^{L}G\to G{L}_d$, which improves the result of \cite{Min25} which says the morphism is representable by locally Noetherian formal algebraic stacks.

28 pages