Yoshida - Non-abelian Lubin-Tate 1

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 $W = \mathcal{O}_{\widehat{K^{ur}}}$-algebras where the structure morphism $W \to A$ is local and induces an isomorphism $$W/\pi \xrightarrow{~\sim~} A/\mathfrak{m}_A$$
• We can thus base change formal $\mathcal{O}_K$-module over $A$ to formal $\mathcal{O}_K$-module over $\overline{\mathbb{F}}_q$ 
• The base change above is called reduction and is denoted by $\overline{\Sigma}$
• The height of formal $\mathcal{O}_K$-module over $A$ is defined by the height of its reduction.
• Given the unique (up to isomorphism) formal $\mathcal{O}_K$-module $\Sigma_n$ over $\overline{\mathbb{F}}_q$, the deformation of $\Sigma_n$ over $A$ is a pair $(\Sigma, i)$ where $i : \Sigma_n \to \overline{\Sigma}$ is an isomorphism of formal $\mathcal{O}_K$-modules over $\overline{\mathbb{F}}_q$. The isomorphism between deformations are formal $\mathcal{O}_K$-isomorphisms with the commuting properties for $i$.
• $\mathcal{F}_0$ is a functor from $\mathcal{C}$ to the category of sets defined by sending $A$ to the isomorphism classes of deformations of $\Sigma_n$ over $A$.
• Proposition (Representability of $\mathcal{F}_0$, [Dr])
• $\mathcal{F}_0$ is represented by the formal power series ring of $n-1$ variables
  $$A_0 = W[[T_1, \ldots, T_{n-1}]]$$
  over $W$. Denote the universal formal $\mathcal{O}_K$-module over $A_0$ by $\widetilde{\Sigma}_n$.

2.3 Deformation with Drinfeld level structure

• For a formal $\mathcal{O}_K$-module $\Sigma$ over $A \in \mathcal{C}$, let $\mathfrak{m}$ be the maximal ideal of $A$. Then $\mathfrak{m}$ becomes a $\mathcal{O}_K$-module by $$x +_\Sigma y := F(x,y) \qquad a \cdot_\Sigma x := [a](x)$$ for $a \in \mathcal{O}_K$. Homomorphism $\Sigma \to \Sigma'$ induces a homomorphism $\mathfrak{m}_\Sigma \to \mathfrak{m}_{\Sigma'}$.
• Now fix a formal $\mathcal{O}_K$-module $\Sigma$ of height $n$ over $A$ and an integer $m \geq 1$. A Drinfeld level $\pi^m$-structure on $\Sigma$ is a $\mathcal{O}_K$-module homomorphism $$\varphi: (\pi^{-m} \mathcal{O}_K/\mathcal{O}_K)^n \to \mathfrak{m}_\Sigma$$ satisfying the divisibility: $$\left . \prod_{x \in (\pi^{-m} \mathcal{O}_K/\mathcal{O}_K)^n} (X - \varphi(x)) ~ \right | ~ [\pi^m](X)$$ in $A[[X]]$. Denote the LHS by $P_\varphi(X)$.
• If $\{ e_1, \ldots, e_n \}$ is the standard basis of $(\pi^{-m} \mathcal{O}_K/\mathcal{O}_K)^n$, then $\{ \varphi(e_i) \}$ is called the formal parameters of the level $\pi^m$-structure.
• By the condition of Drinfeld level $\pi^m$-structure, we have the equality $$P_\varphi(X) = U(X) [\pi^m](X)  \text{ for some }  U(X) \in A[[X]]$$
  Since by definition, both $P_\varphi(X) \equiv [\pi^m](X) \equiv X^{nm} ~(\mathrm{mod}~\mathfrak{m})$. Hence the equation becomes $X^{nm} \equiv P_\varphi(X) X^{nm} ~(\mathrm{mod}~\mathfrak{m})$. Comparing the $X^{nm}$-term, we see that $u_\Sigma$, the constant term of $U(X)$, lies in $1 + \mathfrak{m}$.
• A deformation of $\Sigma_n$ with Drinfeld level $\pi^m$-structure over $A$ is a triple $(\Sigma, i, \varphi)$ where $(\Sigma, i)$ is the usual deformation over $A$ and $\varphi$ is the Drinfeld level $\pi^m$-structure. Isomorphism between deformation with Drinfeld level structure is a morphism of deformation with the obvious commuting property for $\varphi$.
• Define the functor $\mathcal{F}_m : \mathcal{C} \to (\mathrm{Set})$ defined by $A \mapsto$ the isomorphism classes of deformations of $\Sigma_n$ with Drinfeld level $\pi^m$-structure over $A$.
• Proposition (Representability of $\mathcal{F}_m$, [Dr])
• For every integer $m \geq 1$, $\mathcal{F}_m$ is represented by $A_m$, an $n$-dimensional regular local ring.
• If $\mathcal{F}_m \to \mathcal{F}_0$ is the forgetting morphism of functors and $A_0 \to A_m$ is its representing local $W$-algebra homomorphism, then $A_0 \to A_m$ is finite and flat
• The universal object over $A_m$ is a level $\pi^m$-structure $\varphi_m$ on $\widetilde{\Sigma}_n \otimes_{A_0} A_m$. $\varphi_m$ is called the universal level $\pi^m$-structure.
•  The formal parameters $X_1, \ldots, X_n$ of the universal level $\pi^m$-structure gives the regular local parameters of $A_m$.
• The finite flat morphism $A_0 \to A_m$ induces a finite flat covering $\mathrm{Spec}~A_m \to \mathrm{Spec}~A_0$. This is, in fact, a Galois covering.

2.4 Realization as a complete local ring of a Shimura Variety

• We have some Shimura variety. This Shimura variety has a proper flat integral model $X_{U^p, m}$ over $\mathrm{Spec}~\mathcal{O}_{F,w}$ such that $U^p \subset G(\mathbb{A}^{\infty, p})$ and $m=(m_1, \ldots, m_r)$ is a multiindex of natural numbers. If we use some theorem from Harris-Taylor, we can show that the deformation space $\mathrm{Spec}~A_m$ is isomorphic to the completion of the strict local ring of $X_{U^p, m}$ at a closed point $s$ where $h(s) = 0$, i.e. the supersingular point.

Questions:

• (Paragraph after Proposition 2.2) (i) and (iii) seems to contradict each other. I think it should be $$[\pi](X) = \pi X + X^{q^n}$$
• What is the universal formal $\mathcal{O}_K$-module over $A_0$? The universal formal module always exists? 
• What do I need to know about the structure of $\pi^{-m}\mathcal{O}_K/\mathcal{O}_K$?
• Why is $GL_n(\mathcal{O}_K/\pi^m)$ a Galois group? How does this define an automorphism of $\mathrm{Spec}~A_m$?
• What is a strict local ring? Why does the stalk(?) of $X_{U^p, m}$ at a closed point $s$ with h(s) (which I am assuming is the etale height of the corresponding Barsotti-Tate group) equal to 0 same thing as being a supersingular point?