Algorithmic universal étale $(\varphi,Γ)$-modules

Abstract

Let $F/\mathbb{Q}_p$ be a finite extension and let $\breve{G}$ be a connected reductive group with connected center satisfying $p>h_{\breve{G}}$ (the Coxeter number). We write down explicit polynomial equations for the reduced Emerton-Gee stacks (the Borel and the twisted Borel versions that jointly cover the entire $\mathcal{X}_{\breve{G},\mathrm{red}}^{\mathrm{EG}}$), by introducing the mod $p$ Weil-Deligne stacks capturing derived structures of $(\varphi, Γ)$-modules in reduced families. This allows us to algorithmically compute the set of irreducible components of $\mathcal{X}_{\breve{G},\mathrm{red}}^{\mathrm{EG}}$, thereby establishing its equidimensionality, and the existence of crystalline lifts by showing the total number of irreducible components equals the number of (mod $p$) crystalline (or potentially semistable) components. The last step of the arguments is to interpolate the rigid analytic Weil-Deligne stacks and the mod $p$ Weil-Deligne stacks to deduce the number of potentially semistable components. An initial implementation of the algorithms based on Gröbner basis establishes the existence of crystalline lifts in the $\breve{G}=\mathrm{F}_4$ case for all $F/\mathbb{Q}_p$, and in the remaining $\breve{G}=\mathrm{E}_6, \mathrm{E}_7$, and $\mathrm{E}_8$ cases for all but finitely many $F/\mathbb{Q}_p$.

Disclosure

“l discussions, especially Yu for explaining derived algebraic geometry to the author. 1.5. Use of AI. The original manuscript and the original github.com/mocham/AlgEG implementation were written entirely by human. Later, the author used DeepSeek Pro V4 and ChatGPT 5.6 Sol to fix typos and refractor the source code; the refractoring is done mainly by a DeepSeek agent and the verification/testing is done by a ChatGPT agent; the appendices are generated by a ChatGPT agent from the re”

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2026-08-23T21:14:05.655979Z

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