论文标题
有限的德林菲尔德周期域的压缩为蕨类植物的模量空间
Compactification of the finite Drinfeld period domain as a moduli space of ferns
论文作者
论文摘要
令$ \ mathbb {f} _q $为有限字段，带有$ q $元素，让$ v $为$ \ m athbb {f} _q $ dimension $ n> 0 $的矢量空间。令$ω_v$为$ \ mathbb {f} _Q $的Drinfeld周期域。这是$ \ mathbb {f} _q $的有限类型的仿射方案，其基本更改为$ \ mathbb {f} _q（t）$是drinfeld $ \ mathbb {f} _q [t] $ - 具有级别的$（T）$（t）$结构$ clubs $ n $ n $ $ n $的模量。在本文中，我们对Pink和Schieder的平滑压实$ b_v $ $ω_v$进行了新的模块化解释。让$ \ hat v $为新符号$ \ infty $的集合$ v \ cup \ {\ infty \} $。我们定义了$ v $ - fern的概念，而不是$ \ mathbb {f} _q $ -scheme $ s $，它由稳定的$ \ hat v $ - $ 0 $ 0 $ 0 $ s $ s $ S $的曲线组成。我们的主要结果是该方案$ b_v $代表函数，将$ \ mathbb {f} _q $ -scheme $ s $与$ s $ $ s $的同构类别类别相关联。因此，$ v $ - ferns over $ \ mathbb {f} _q（t）$ - 方案可以被视为drinfeld $ \ mathbb {f} _q [t] $ - 具有级别$（t）$结构和等级$ n $的模块的概括。为了证明此定理，我们在$ b_v $上构建了一个明确的通用$ v $ -fern。然后，我们证明，在方案$ s $上的任何$ v $ - fern都决定了唯一的形态$ s \ to b_v $，具体取决于其同构类别，而$ v $ - fern对沿这种形态的punidence $ v $ - fern的回调是同构的。我们还提供了几个涉及$ V $ - Ferns的功能构造，其中一些用于证明主要结果。这些构造对应于$ \ mathbb {f} _q $的Drinfeld周期域的各种模块化压实之间的形态。我们明确描述了这些形态。
Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $V$ be a vector space over $\mathbb{F}_q$ of dimension $n>0$. Let $Ω_V$ be the Drinfeld period domain over $\mathbb{F}_q$. This is an affine scheme of finite type over $\mathbb{F}_q$, and its base change to $\mathbb{F}_q(t)$ is the moduli space of Drinfeld $\mathbb{F}_q[t]$-modules with level $(t)$ structure and rank $n$. In this thesis, we give a new modular interpretation to Pink and Schieder's smooth compactification $B_V$ of $Ω_V$. Let $\hat V$ be the set $V\cup\{\infty\}$ for a new symbol $\infty$. We define the notion of a $V$-fern over an $\mathbb{F}_q$-scheme $S$, which consists of a stable $\hat V$-marked curve of genus $0$ over $S$ endowed with a certain action of the finite group $V\rtimes \mathbb{F}_q^\times$. Our main result is that the scheme $B_V$ represents the functor that associates an $\mathbb{F}_q$-scheme $S$ to the set of isomorphism classes of $V$-ferns over $S$. Thus $V$-ferns over $\mathbb{F}_q(t)$-schemes can be regarded as generalizations of Drinfeld $\mathbb{F}_q[t]$-modules with level $(t)$ structure and rank $n$. To prove this theorem, we construct an explicit universal $V$-fern over $B_V$. We then show that any $V$-fern over a scheme $S$ determines a unique morphism $S\to B_V$, depending only its isomorphism class, and that the $V$-fern is isomorphic to the pullback of the universal $V$-fern along this morphism. We also give several functorial constructions involving $V$-ferns, some of which are used to prove the main result. These constructions correspond to morphisms between various modular compactifications of Drinfeld period domains over $\mathbb{F}_q$. We describe these morphisms explicitly.
