Library of Mathexandria

Now we would like to study the special fiber of $X:=\mathrm{A}$ over $S:=\mathrm{Spec}~W$. We denote the special fiber by $X_s:= X \times_S \mathrm{Spec}~\overline{k}$, i.e. the fiber of the special point (the maximal ideal). Sine this is simply just reducing mod $\pi$, combining with the result from the previous post where we computed the explicit formula for $A$, we obtain that \[ X_s = \mathrm{Spec}~ \overline{k}[[\widetilde{X}_1, \ldots, \widetilde{X}_n]]/ \left ( \prod_{\underline{a} \in k^n \backslash \{ 0 \}} (P_{\underline{a}} ~\mathrm{mod}~ \pi ) \right ) \] Let $Y_{\underline{a}}$ to be the closed subscheme of $X_s$ defined by $P_{\underline{a}} ~\mathrm{mod}~\pi = 0$. Equivalently, this is simply the closed subscheme of $X$ defined by $P_{\underline{a}} = 0$. It can be shown that $Y_{\underline{a}} = Y_{\underline{a'}}$ if and only if $a$ is a scalar multiple of $a'$ by some element in $k^\times$. Therefore, we can label the irreducible component of $X_s...

We come back to the original paper. $A:=A_1$ is the representing algebra of the deformation functor $\mathcal{F}_1: \mathcal{C} \to (\mathrm{Set})$ that sends local $W$-algebras with additional properties to the set of isomorphism classes of deformations with level-$\pi$-structures. By the theorem of Drinfeld, we have the universal deformation and universal level $\pi$-structure \[ \widehat{\Sigma} :=\widetilde{\Sigma} \otimes_{A_0} A_1 \text{ and } \varphi:= \varphi_1 : (\pi^{-1} \mathcal{O}_K/\mathcal{O})^n \to \mathfrak{m}_{\widehat{\Sigma}} \]  Then $X_i:=\varphi(e_i)$ where $e_i$ are the standard basis of $(\pi^{-1} \mathcal{O}_K/\mathcal{O})^n$. This forms a system of local parameters of $A$. One of the conditions of level $\pi$-structure is that  \[ P_\varphi(T) { \Large| } [\pi](T) \] with $P_\varphi(T)U(T) = [\pi](T)$ (*) $U(T)$ has constant term $u_{\widehat{\Sigma}} \in 1 + \mathfrak{m}$. \[ P_\varphi(T) = \prod_{x \in (\pi^{-1} \mathcal{O}_K/\mathcal{O}...

Some remarks. Let $f:X \to Y$ be a morphism of (integral) schemes. Then we say that $f$ is a Galois covering  if $K(X)/K(Y)$ is a Galois extension. $A_0$ is clearly a domain and $A_m$ is a regular local ring, then by Matsumura, Theorem 14.3, we know that $A_m$ is a domain. Therefore $\mathrm{Spec}~A_m \to \mathrm{Spec}~A_0$ is a morphism of integral schemes. According to the post in mathSE, $Y$ becomes a quotient of $X$ by the group of deck transformations of $f$ which turns out to be the Galois group.  According to the Wiki article , discrete valuation ring is a regular ring with dimension 1. The converse is also true. Therefore $A_0:=W[[X_1, \ldots, X_{n-1}]]$ is a regular ring with dimension $n$. Review on representable functor A functor $\mathcal{F}: \mathcal{C} \to (\mathrm{Set})$ is called representable if there exists $A \in \mathcal{C}$ such that $\mathcal{F}$ is naturally isomorphic to $Hom(A,-)$ (or $Hom(-,A)$ for contravariant functor). A pair $(...

This will be a series of personal notes on the paper of Yoshida, "On Non-abelian Lubin-Tate Theory via Vanishing Cycles". The general format would be to have outlines in the first part and (stupid) questions in the second part. Also, in the middle, I will put background materials. Hopefully, most of the questions will be answered by the end of the month. Excursion. Moduli Space 1. Moduli Space of Elliptic Curves  \( \require{AMScd} \) Consider the category \( (\mathrm{Ell}) \) where the objects are elliptic curves over an arbitrary base scheme \[ \begin{CD} E \\ @VV{\pi}V \\ S \end{CD} \] and the morphisms are the Cartesian squares \[ \begin{CD} E_1 @>{a}>> E \\ @V{\pi_1}VV @VV{\pi}V \\ S_1 @>{f}>> S \end{CD} \] The above Cartesian diagram induces an isomorphism \[ E_1 \xrightarrow{~(\alpha, \pi_1)~} E \times_S S_1 \] \( (\mathrm{Ell}) \) is called the modular stack of Deligne-Rapoport . A contravariant functor \( \mathcal{P} ...

This will be a series of personal notes on the paper of Yoshida, "On Non-abelian Lubin-Tate Theory via Vanishing Cycles". The general format would be to have outlines in the first part and (stupid) questions in the second part. Also, in the middle, I will put background materials. Hopefully, most of the questions will be answered by the end of the month. Current Reference: Drinfeld, "Elliptic Modules". 1. Formal Modules The definition of a formal group and a homomorphism coincide with the one with Yoshida.  There is a canonical homomorphism \( D: \mathrm{End}(F) \to B \) where \( F \) is a formal group over a ring \( B \). If \( \phi \) is an endomorphism of \( F \), then \( D(\phi) = \phi'(0) \). This is a homomorphism as \( D(\phi \circ \psi) = \phi'(\psi(0)) \cdot \psi'(0) \), but as \( \psi \in (X) \subset B[[X]] \), we conclude that \( D(\phi \circ \psi) = \phi'(0) \psi'(0) \). (Example) \( F(X,Y) = X+Y \) is a f...

This will be a series of personal notes on the paper of Yoshida, "On Non-abelian Lubin-Tate Theory via Vanishing Cycles". The general format would be to have outlines in the first part and (stupid) questions in the second part. Also, in the middle, I will put background materials. Hopefully, most of the questions will be answered by the end of the month. 3. The Level \( \pi \) Deformation Space 3.1 The equation of the space We would like to study \( X = \mathrm{Spec}~A_1 \) where \( A_1 \) is the deformation ring defined from the previous post. Then \( X \) is a regular flat scheme over \( S = \mathrm{Spec}~W \) of relative dimension \( n-1 \) with a (formally) smooth generic fiber. Notation-wise, we shorten \( \widetilde{\Sigma}_n \) as simply \( \widetilde{\Sigma} \). Similarly, we shorten \( A_1 \) as simply \( A \). We denote the generators of \( \mathfrak{m} \), the maximal ideal of \( A\), by \( X_1, \ldots, X_n \), i.e., \( \mathfrak{m} = (X_1, \ld...

This will be a series of personal notes on the paper of Yoshida, "On Non-abelian Lubin-Tate Theory via Vanishing Cycles". The general format would be to have outlines in the first part and (stupid) questions in the second part. Also, in the middle, I will put background materials. Hopefully, most of the questions will be answered by the end of the month. 2. Review on the Moduli Spaces of Formal \( \mathcal{O}_K \)-modules 2.1 Formal \( \mathcal{O}_K \)-modules Definition of Formal \(\mathcal{O}_K \)-modules, homomorphism/endomorphism, base change Additive group is a formal \(\mathcal{O}_K \)-module For any formal \(\mathcal{O}_K \)-module (over \( \overline{\mathbb{F}}_q \) ) non-isomorphic to the additive group, there exists a unique height. We can "normalize" the formal \(\mathcal{O}_K \)-module to satisfy certain properties. 2.2  Deformation of formal \( \mathcal{O}_K \)-modules Category \( \mathcal{C} \) of local Noetherian \( W = \mat...