Yoshida - Non-abelian Lubin-Tate 4

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 $\mathcal{P} : (\mathrm{Ell}) \to (\mathrm{Set})$ is called a moduli problem for Elliptic Curves.
• Given $E/S$, the set $\mathcal{P}(E/S)$ is called a level-$\mathcal{P}$-structure.
• The moduli problem $\mathcal{P}$ is called relatively representable over $(\mathrm{Ell})$ if for every elliptic curve $E/S$, the functor $(\mathrm{Sch/S}) \to  (\mathrm{Set})$ defined by $T \mapsto \mathcal{P}(E_T/T)$ is representable. We denote such $S$-scheme by $\mathcal{P}_{E/S}$.
• The moduli problem $\mathcal{P}$ is representable if it is representable as a functor on $(\mathrm{Ell})$.  We denote the representing elliptic curve $\mathbf{E}/\mathcal{M}(\mathcal{P})$.
• Therefore, we have an isomorphism
  $$\mathcal{P}(E/S) \cong Hom_{(\mathrm{Ell})}(E/S, \mathbf{E}/\mathcal{M}(\mathcal{P}))$$
• $\mathcal{M}(\mathcal{P})$ represents the functor $(\mathrm{Sch}) \to (\mathrm{Set})$ defined by $S \mapsto$
  
  isomorphism classes of pairs $(E/S, \alpha)$ with $E$ an elliptic curve over $S$ and $\alpha \in \mathcal{P}(E/S)$ a level-$\mathcal{P}$-structure on $E/S$
• Converse is also true. If $\mathcal{M}(\mathcal{P})$ is representable by a scheme $\mathcal{M}(\mathcal{P})$ with universal object $(\mathbf{E}/\mathcal{M}(\mathcal{P}), \alpha)$ and if $\mathcal{P}$ is rigid, then $\mathcal{P}$ is represented by the object $\mathbf{E}/\mathcal{M}(\mathcal{P})$ of $(\mathrm{Ell})$.

2. Universal Object

• Let $U: \mathcal{C} \to \mathcal{D}$ be a functor. Let $X \in \mathcal{D}$.  A initial morphism from$X$ to $U$ is a pair $(A, \alpha)$ where $A$ is an object of $\mathcal{C}$ and $\phi: X \to U(A)$ with the following universal property: whenever $Y$ is an object of $\mathcal{C}$ and $f: X \to U(Y)$ such that $f, \phi, \alpha$ commute in the obvious way. (MathJAX does not support diagonal arrows.)
• As of now, I have no idea how the two connects. I want to know whether or not this definition follows, and there exists a morphism $\alpha: X \to \mathcal{P}(\mathbf{E}/\mathcal{M}(\mathcal{P}))$ with some set $X$ such that $(\mathbf{E}/\mathcal{M}(\mathcal{P}), \alpha)$ is a universal object of $\mathcal{P} : (\mathrm{Ell}) \to (\mathrm{Set})$.