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 \( \mathcal{P} : (\mathrm{Ell}) \to (\mathrm{Set}) \).

Comments

Post a Comment

Popular posts from this blog

The topology of the p-adic numbers 3

Let \( G \) be a topological space and let \( H \) be a subspace of \( G \) endowed with the subspace topology. Then \( H \times H \) endowed with the product topology is same as the subspace topology of \( H \times H \subset G \times G \) with \(G \times G \) endowed with the product topology due to: \[ (U_1 \cap H) \times (U_2 \cap H) = (U_1 \times U_2) \cap (H \times H) \] Let \( G \) be a topological group with a subgroup \( H \). We would like to show that \( H \) endowed with the subspace topology is also a topological group. Let \( p_G \) denote the product map. in \( G \) and \( p_H \) denote the product map in \( H \). We want to show that \( p_H \) is continuous. Let \( U \) be an open set in \( H \), then \(U = U_G \cap H \) for some open set \( U_G \) in \( G \). \[ \begin{array}{lcl} p_H^{-1}(U) & = & \{ (h_1, h_2) \in H \times H ~|~ h_1h_2 \in U \} \\ & = & \{ (g_1, g_2) \in G \times G ~|~ g_1g_2 \in U_G \} \cap H \times H \\ & = & p_G^{-1}(U_G) \...
Read more
Question. when does a morphism of schemes send a closed point to a closed point?  1) Integral Morphisms This amounts to saying that $B/A$ integral extension, then $B$ is a field if and only if $A$ is a field. 2) Projective => Proper => Closed => preserves closed point. Reference. mathSE post Let $X$ be a scheme of finite type over a finite (or algebraically closed) field $k$. Let $K$ be a finite extension of $\mathbb{Q}_p$ with residue field $k$.  We would like to define $D_c^b(X, \overline{\mathbb{Q}}_l)$. 1) $D_c^b(X, \mathcal{O}_r)$ where $\mathcal{O}_r$ is $\mathcal{O}/\pi^r \mathcal{O}$.
Read more

Max-Spec vs Spec

This is an article on why using Max-Spec and Spec do not matter much if the ring $A$ is a Jacobson ring.  Definition. A ring $A$ is a Jacobson ring if every prime ideal of $A$ is an intersection of maximal ideals. In other words, \[ \mathrm{Nil}(A) = \bigcap_{\mathfrak{p}} \mathfrak{p} = \bigcap_{\mathfrak{m}} \mathfrak{m} = \mathrm{Jac}(A) \]  Proposition.  Let $X =\mathrm{Spec}(A)$ and $X_0 = \mathrm{MaxSpec}(A)$, i.e. the set of closed points of $X$. The $X_0$ is a very dense subset of $X$. We show that some topological properties are preserved. Let's first describe what it means for $\mathrm{Spec}(A)$ to be disconnected. By this means that there exist disjoint closed subsets $V(I)$ and $V(J)$ that cover $\mathrm{Spec}(A)$. In other words, 1) $V((0))=\mathrm{Spec}(A) = V(I) \cup V(J) = V(I \cap J)$.  2) $V((1)) = \varnothing = V(I) \cap V(J) = V(I + J)$. This is same as saying that 1') $\sqrt{I \cap J} = \sqrt{(0)} = \mathrm{Nil}(A...
Read more