Yoshida - Non-abelian Lubin-Tate 6


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 $u_{\widehat{\Sigma}} \in 1 + \mathfrak{m}$.
  • \[ P_\varphi(T) = \prod_{x \in (\pi^{-1} \mathcal{O}_K/\mathcal{O}_K)^n} (T - \varphi(x)) \]
    • Here $0 \in (\pi^{-1} \mathcal{O}_K/\mathcal{O}_K)^n$ yields $\varphi(0)$, so, in fact, we have
      \[ P_\varphi(T) = T \cdot \prod_{x \in (\pi^{-1} \mathcal{O}_K/\mathcal{O}_K)^n \backslash \{ 0 \}} (T - \varphi(x)) \]
    • Therefore $P_\varphi(T)$ has no constant term and has $\prod \varphi(x)$ as its coefficient for $T$. 
    • There is no sign because $|\pi^{-1}\mathcal{O}_K/\mathcal{O}_K| = q$ odd, so $q^n -1$ is even.
  • By $\mathcal{O}_K \to W:= \widehat{\mathcal{O}_{K^{ur}}} \to A$, we know that $A$ is also a $\mathcal{O}_K$-algebra. Then $[\pi](T) = \pi T$by the previous remark shows that
    \[ \pi = u_{\widehat{\Sigma}} \cdot \prod_{x \in (\pi^{-1} \mathcal{O}_K/\mathcal{O}_K)^n \backslash \{ 0 \}} \varphi(x) \]
  • Then I think the fact that $(\pi^{-1} \mathcal{O}_K/ \mathcal{O}_K)^n$ is a $k$-vector space, we get $x= a_1e_1 + \cdots + a_ne_n$ with $a_i \in k$. Combined with the definition of formal parameters, we get
    \[ \pi = u_{\widehat{\Sigma}} \cdot \prod_{\underline{a} \in k^n \backslash \{ 0 \}} \varphi(a_1 e_1 + \cdots + a_n e_n) = u_{\widehat{\Sigma}} \cdot \prod_{\underline{a} \in k^n \backslash \{ 0 \}} \varphi([a_1](X_1)+_{\widehat{\Sigma}} \cdots +_{\widehat{\Sigma}} [a_n](X_n)) \]


  • Still need to figure out where $W[[\widetilde{X}_1, \ldots, \widetilde{X}_n]]$ came from.
  • From the map $W[[\widetilde{X}_1, \ldots, \widetilde{X}_n]] \to A$ which is surjective.  We would like to describe $I$ that makes $A \cong W[[\widetilde{X}_1, \ldots, \widetilde{X}_n]]/I$. Pick $t' \in I \backslash \widetilde{\mathfrak{m}}$. We can show that there exists an isomorphism between $(t')/\widetilde{\mathfrak{m}}(t') \to I/\widetilde{\mathfrak{m}}I$ by proving that they are injective and both 1-dimensional $k$-vector spaces. Then applying the Nakayama, we see that $(t') = I$.
  • How do we find such $t'$? 
  • By the definition of the deformation space, we can find $f$ such that the following diagram commutes
    \[ \begin{array}{ccc} W[[\widetilde{X}_1, \ldots, \widetilde{X}_n]] & \rightarrow & A \\ & \nwarrow & \uparrow \\ & & A_0=W[[T_1, \ldots, T_{n-1}]] \end{array} \]
    The existence of $f$ is ensured by the formal smoothness of $A_0$ over $W$. 
  • Using such $f$, we can lift $\widetilde{\Sigma}$ to $W[[\widetilde{X}_1, \ldots, \widetilde{X}_n]]$ by $\widetilde{\Sigma} \otimes_{A_0, f} W[[\widetilde{X}_1, \ldots, \widetilde{X}_n]]$. Denote this by $\widetilde{\Sigma}^+$. Also for $a \in k$, we fix a lift $\widetilde{a} \in \mathcal{O}_K$. One of the standard choice would be $\mu_{q-1}$. Now we can define
    \[ P_{\underline{a}}(\widetilde{X}_1, \ldots, \widetilde{X}_n) = [\widetilde{a}_1](\widetilde{X}_1) +_{\widetilde{\Sigma}^+} \cdots +_{\widetilde{\Sigma}^+} [\widetilde{a}_n](\widetilde{X}_n) \]
    where $\underline{a} =(a_1, \ldots, a_n)$.
  • Define $u$ to be the lift of $u_{\widehat{\Sigma}} \in A$ to $W[[\widetilde{X}_1, \ldots, \widetilde{X}_n]]$, then we can define the formal power series,
    \[ P(\widetilde{X}_1, \ldots, \widetilde{X}_n) = u \cdot \prod_{\underline{a} \in k^n \backslash \{ 0 \}} P_{\underline{a}}(\widetilde{X}_1, \ldots, \widetilde{X}_n) \] 
  • Then we can now describe $A$ as follows:
    \[ W[[\widetilde{X}_1, \ldots, \widetilde{X}_n]]/(P-\pi) \cong A \]
  • For arbitrary $m$, we can generalize by letting
    \[ P = u \cdot \prod_{\underline{a} \in (\mathcal{O}_K/\pi^m)^n \backslash (\pi \mathcal{O}_K/\pi^m)^n} P_{\underline{a}} \]

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