论文标题
$ p $ - adic模块化表单上的一项单一圆圈动作
A unipotent circle action on $p$-adic modular forms
论文作者
论文摘要
在提出了彼得·索尔兹（Peter Scholze）的建议之后，我们构建了$ \ hat {\ mathbb {g}} _ m $在katz moduli问题上的动作，这是$ p $ - addic modular曲线的普通基因座的profinite-étale封面，其功能环的空间是$ p $ p $ dadic modular函数的空间。此操作是圆圈组的全局$ s^1 $的本地，$ p $ -Adic的类似物，在$ \ Mathbb {c} $上的晶格 - unstable locus上。要构建$ \ hat {\ mathbb {g}} _ m $ - action，我们在（大）普通的igusa caraiani-scholze品种上降低了一个较大群体的模量理论动作。我们在本地扩展上明确计算操作，并通过简单的cuspidal和serre-tate坐标$ q $的简单乘法给出了操作；一路上，我们还证明了DWork的方程式$τ= \ log Q $的自然概括，用于$ \ Mathbb {q} _p/\ Mathbb {z} _p $ by $μ_p$ by $μ_{p^\ infty} $有效于非Artininian基础。最后，我们给出了一个直接的论点（无需吸引本地扩展），以表明$ \ hat {\ mathbb {g}} _ m $的动作集成了差异操作员$θ$来自高斯 - 曼宁连接和单位根部拆分，并解释对Eisenstein Measitures和$ p $ -Adic $ l $ l $ l $ frunctions的申请。
Following a suggestion of Peter Scholze, we construct an action of $\hat{\mathbb{G}}_m$ on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the $p$-adic modular curve whose ring of functions is Serre's space of $p$-adic modular functions. This action is a local, $p$-adic analog of a global, archimedean action of the circle group $S^1$ on the lattice-unstable locus of the modular curve over $\mathbb{C}$. To construct the $\hat{\mathbb{G}}_m$-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates $q$; along the way we also prove a natural generalization of Dwork's equation $τ=\log q$ for extensions of $\mathbb{Q}_p/\mathbb{Z}_p$ by $μ_{p^\infty}$ valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of $\hat{\mathbb{G}}_m$ integrates the differential operator $θ$ coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and $p$-adic $L$-functions.
