Moduli stacks of generalized phi-modules
Zhongyipan Lin
TL;DR
The paper develops a finite-presentation, height-theoretic framework for moduli of generalized $\varphi$-modules with Langlands data, proving that change-of-group maps between Emerton–Gee stacks are relatively representable by algebraic stacks of finite presentation over $\operatorname{Spf}\mathbf{Z}_p$ for any embedding into $\operatorname{GL}_d$, and establishing a Shapiro-type equivalence among base fields. Central to the approach is a height theory for $\varphi$-modules, yielding $\mathcal{R}_{{^{L}\!G}}^{\le h}$ whose colimit recovers $\mathcal{R}_{{^{L}\!G}}$, and a universal Kisin lattice family over the corresponding Kisin stacks, enabling descent from loop groups to finite-type moduli. The authors also develop parabolic stacks $\mathcal{X}_{{^{L}\!P}}$ via Herr complexes and tautological torsors, proving these are Ind-algebraic with controlled transition maps and vanishing loci for higher cup products. Together, these results provide a robust finiteness framework for moduli of generalized $\varphi$-modules with reductive structure, opening pathways for constructing and studying potentially semistable substacks and their interactions under group-restriction and Shapiro-type phenomena.
Abstract
Let $F$ be an arbitrary $p$-adic field and let $G$ be an arbitrary reductive group over $F$ with Langlands dual group $^LG$. We show that the change-of-group morphism of Emerton-Gee stacks $\mathcal{X}_{^LG}\to\mathcal{X}_{GL_d}$ is relatively representable by algebraic stacks of finite presentation over $\operatorname{Spf}\mathbf{Z}_p$ for any embedding $^LG\to GL_d$, which improves the result of \cite{Min25} which says the morphism is representable by locally Noetherian formal algebraic stacks.