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?  

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