AI 中文总结
针对满足$p>h_{\breve{G}}$的连通约化群$\breve{G}$，通过引入模$p$韦伊-德莱恩栈算法计算约化Emerton-Gee栈的不可约分支，确立其等维性与晶态提升存在性，且已在$\text{F}_4$等情形验证该算法。
AI 中文摘要
设$F/\boldsymbol{\text{Q}}_p$为有限扩张，$\breve{G}$为满足$p>h_{\breve{G}}$（$h_{\breve{G}}$为Coxeter数）的连通中心的连通约化群。我们通过引入约化族中捕获$(φ,Γ)$-模导出结构的模$p$韦伊-德莱恩栈，写出约化Emerton-Gee栈（共同覆盖整个$\boldsymbol{\text{X}}_{\breve{G},\text{red}}^{\text{EG}}$的Borel版本与扭曲Borel版本）的显式多项式方程。这使我们能算法计算$\boldsymbol{\text{X}}_{\breve{G},\text{red}}^{\text{EG}}$的不可约分支集合，从而确立其等维性，并通过不可约分支总数等于（模$p$）晶态（或潜在半稳定）分支数证明晶态提升的存在性。论证最后一步是插值刚性解析韦伊-德莱恩栈与模$p$韦伊-德莱恩栈以推导潜在半稳定分支数。基于Gröbner基的算法初始实现，确立了$\breve{G}=\text{F}_4$时所有$F/\boldsymbol{\text{Q}}_p$的晶态提升存在性，以及$\breve{G}=\text{E}_6、\text{E}_7、\text{E}_8$时除有限个$F/\boldsymbol{\text{Q}}_p$外的晶态提升存在性。
英文摘要
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 $(φ, Γ)$-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 $F/\mathbb{Q}_p$.
Comments68 pages
