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An infinite dimensional Hodge-Tate theory. (English) Zbl 0786.11067
Let $$K$$ be a $$p$$-adic field, $$\overline{K}$$ an algebraic closure of $$K$$, and $$\mathbb{C}$$ the completion of $$\overline{K}$$ with respect to the absolute norm $$|\;|$$. Let $${\mathcal G}= \text{Gal}(\overline{K}/K)$$ be the absolute Galois group. Let $$\chi: {\mathcal G}\to \mathbb{Q}_ p^*$$ be the $$p$$- cyclotomic character. We set up some notation: $${\mathcal H}= \text{Ker} \chi$$, $$\Gamma= \text{Gal} (\overline{K}^{\mathcal H} /K)$$, $$\Gamma_ 0$$ = the maximal pro-$$p$$-subgroup of $$\Gamma$$, which is $$\simeq\mathbb{Z}_ p$$, $$\Gamma_ n= \Gamma_ 0^{p^ n}$$, and $${\mathcal G}_ n$$ = the inverse image in $${\mathcal G}$$ of $$\Gamma_ n$$. If $${\mathcal G}'$$ is an open subgroup of $${\mathcal G}$$, the corresponding objects $${\mathcal H}'$$, $$\Gamma'$$, $$\Gamma_ 0'$$, $$\Gamma_ n'$$, $${\mathcal G}_ n'$$, etc. are similarly defined.
Let $$V$$ be a finite dimensional $$K$$-vector space, and $$X$$ the $$\mathbb{C}$$- vector space $$V\otimes_ K \mathbb{C}$$ on with $${\mathcal G}$$ acts diagonally. A continuous homomorphism $$\psi: {\mathcal G}\to \operatorname{Aut}_ K V$$ is said to have a Hodge-Tate decomposition if $$X= \sum_{i\in\mathbb{Z}} X(i)$$ where $$X(i)$$ is the $$\mathbb{C}$$-subspace of $$X$$ spanned by all $$x\in X$$ such that $$\sigma(x)=\chi^ i (\sigma)x$$ for all $$\sigma\in{\mathcal G}$$. In [Invent. Math. 62, 89-116 (1980; Zbl 0463.12005)] the author constructed, for arbitrary $$\psi$$, a canonical operator $$\varphi$$ on $$X$$ which has the eigenvalue $$i$$ and the corresponding eigenspace $$X(i)$$ whenever the latter is non-zero. Such an operator $$\varphi$$ is regarded as giving a generalized Hodge-Tate decomposition for any $$\psi$$.
In this paper, the construction of the “canonical” operator $$\varphi$$ is generalized, and along the way, the author also corrects a technical error in the paper [(*) Ann. Math., II. Ser. 127, 647-661 (1988; Zbl 0662.12018)], but even strengthens the results proved therein. Further, the uniqueness of such an operator $$\varphi$$ is proved.
Let $${\mathcal B}$$ be a Banach $$\mathbb{C}$$-algebra on which $${\mathcal G}$$ acts properly, that is,
(1) $${\mathcal G}$$ preserves the norm of $${\mathcal B}$$,
(2) The function $${\mathcal G} \times {\mathcal B}\to {\mathcal B}$$ given by the action of $${\mathcal G}$$ is continuous (for the Krull topology of $${\mathcal G}$$ and the norm topology of $${\mathcal B}$$,
(3) For any open subgroup $${\mathcal G}'$$ of $${\mathcal G}$$, the action of $$\Gamma'$$ on $${\mathcal B}_ \infty'= {\mathcal B}^{{\mathcal H}'}$$ satisfies the following two conditions: (a) For each $$n$$, $${\mathcal B}_ \infty'= {\mathcal B}_ n'\oplus (\gamma_ n'- 1){\mathcal B}_ \infty'$$ where $${\mathcal B}_ n'= ({\mathcal B}_ \infty')^{\Gamma_ n'}$$ and $$\gamma_ n'$$ is a topological generator of $$\Gamma_ n'$$, and (b) the inverse operators $$(\gamma_ n'-1)^{-1}$$ on $$(\gamma_ n'-1){\mathcal B}_ \infty'$$ satisfy a uniform bound $$d$$ (i.e., independent of $$n$$, but which may still depend on $${\mathcal G}'$$.)
(If $${\mathcal R}$$ is a Banach $$K$$-algebra, then $${\mathcal B}= {\mathcal R} \widehat {\otimes}_ K \mathbb{C}$$ is such a Banach $$\mathbb{C}$$-algebra.) Let $${\mathcal B}^*$$ denote the group of units of $${\mathcal B}$$. Let $$u:{\mathcal G}\to {\mathcal B}^*$$ be a semi-linear representation, i.e., a continuous 1- cocycle.
Then an operator $$\varphi\in{\mathcal B}$$ is defined as follows. Note that there is an open subgroup $${\mathcal G}'\subset {\mathcal G}$$ such that $$u({\mathcal G}')\subset {\mathcal B}^ 1$$ where $${\mathcal B}^ 1= \{x\in {\mathcal B}\mid |1-x|<1\}$$. Since $$H^ 1({\mathcal H}',{\mathcal B}^ 1)=1$$ [Prop. 1, in (*) (loc. cit.)], there is a cocycle $$v$$ on $${\mathcal G}/{\mathcal H}'$$ with values in $${\mathcal B}^{{\mathcal H}'}$$ whose inflation is cohomologous to $$u$$ on $${\mathcal G}'$$. Let $$\widehat {K}_ \infty'= \mathbb{C}^{{\mathcal H}'}$$. Then the $$\widehat {K}_ \infty'$$-Banach algebra $${\mathcal B}_ \infty'$$ satisfies the condition of [Prop. 3 in (*)]. From this and the assumption of the proper action of $${\mathcal G}$$, one sees that for sufficiently large $$n$$, there is a continuous homomorphism $$\rho: \Gamma_ n'\to {\mathcal B}_ n^{\prime *}$$ such that $$\rho$$ is cohomologous to the restriction of $$v$$ to $$\Gamma_ n'$$. Thus $$\rho$$ is cohomologous to the restriction of $$u$$ to $${\mathcal G}_ n'$$, that is, there is an element $$m\in{\mathcal B}^*$$ such that $$m^{-1} u(\sigma) \sigma(m)= \rho(\sigma)$$ for all $$\sigma\in {\mathcal G}_ n'$$. Define $\varphi= \lim_{\sigma\to 1} {{m\log \rho(\sigma)m^{-1}} \over {\log\chi(\sigma)}}$ where log is the $$p$$-adic logarithm. It is shown that $$\varphi$$ is well-defined (in fact, when $$\sigma$$ is close to 1, $$\sigma=\gamma^ a$$ with $$a\in\mathbb{Z}_ p$$ where $$\gamma$$ is a topological generator of $$\mathbb{Z}_ p$$, so that the quotient is constant), and depends only on $$u$$ and not on the choice of $$m$$ and $$\rho$$. The uniqueness of $$\varphi$$ is established in this paper.
Furthermore, when $${\mathcal B}= {\mathcal R}\widehat {\otimes}_ K \mathbb{C}$$ where $${\mathcal R}$$ is a Banach $$K$$-algebra and $$K$$ is locally compact, it is shown that the conjugacy class of such an operator in $${\mathcal B}$$ determines the $$\mathbb{C}$$-extension of $$\rho$$, up to local isomorphism (i.e., on an open subgroup of $${\mathcal G})$$. More precisely, the mapping $$u\mapsto \varphi(u)$$ establishes a bijection between the local isomorphism classes $$[u]$$ of semi-linear representations in $${\mathcal B}$$ and the conjugacy classes $$\{b\}$$ of admissible elements $$b\in {\mathcal B}$$. (Here two elements $$b_ 1,b_ 2\in{\mathcal B}$$ are said to be conjugates if $$b_ 2=xb_ 1 x^{-1}$$ for some $$x\in {\mathcal B}^*$$, and $$b\in{\mathcal B}$$ is called admissible if $$b$$ is conjugate to $$b'\in\{b\}$$ such that $$b'\in {\mathcal B}^{{\mathcal G}'}$$ for some open subgroup $${\mathcal G}'$$ of $${\mathcal G}$$.)
The previous construction of the canonical operator $$\varphi$$ is carried out for the following special case of $${\mathcal R}$$. Let $$D$$ be the universal deformation ring of a residual representation as in B. Mazur [Publ., Math. Sci. Res. Inst. 16, 385-437 (1989; Zbl 0714.11076)]. One knows that $$D\simeq {\mathcal O}_ E [[T_ 1,\cdots,T_ \ell]]/I$$ where $$E$$ is a finite extension of $$\mathbb{Q}_ p$$ (contained in $$K$$), and $${\mathcal O}_ E$$ denotes the ring of integers of $$E$$. Let $$\psi_ 0: {\mathcal G}\to \text{GL}_ n(D)$$ be the local restriction of the universal deformation. Let $$\mathbb{C}[[T]]= \mathbb{C}[[T_ 1,\cdots, T_ \ell]]$$. Define a norm $$|\;|_ k$$ on $$\mathbb{C}[[T]]$$ for $$k=(k_ 1,\cdots,k_ \ell)$$ with $$0<k_ i<1$$ by letting $| f(T)|_ k=\sup \{| a_ r T^ r| \text{ for } f(T)=\sum_ r a_ r T^ r\},$ where $$| aT_ 1^{r_ 1}\cdots T_ \ell^{r_ \ell}|=| a| k_ 1^{r_ 1}\cdots k_ \ell^{r_ \ell}$$. Consider a subring of $$\mathbb{C}[[T]]$$ defined by $\mathbb{C}\langle T\rangle_ k= \{f=\sum a_ rT^ r\in \mathbb{C}[[T]]\mid | a_ r T^ r|\to 0\text{ as } r\to\infty\},$ and let $$K\langle T\rangle_ k= K[[T]]\cap \mathbb{C}\langle T\rangle_ k$$. Then $$K\langle T\rangle_ k$$ is a Banach algebra with the norm $$|\;|_ k$$. Let $$J$$ denote the closure of the image of $$I$$ under the embedding $${\mathcal O}_ E[[T]]\hookrightarrow K\langle T\rangle_ k$$ and let $${\mathcal P}= K\langle T\rangle_ k/J$$. Now let $${\mathcal R}$$ be an $$n\times n$$ matrix algebra over $${\mathcal P}$$ with the standard sup norm. Let $$\psi$$ denote the composed map $${\mathcal G}@>\psi_ 0>> \text{GL}_ n(D)\to {\mathcal R}^*$$. Then one can carry out the above construction of $$\varphi(\psi_ \mathbb{C})\in {\mathcal B}= {\mathcal R} \widehat {\otimes}\mathbb{C}$$. If we identify $$K\langle T\rangle_ k \widehat{\otimes} \mathbb{C}$$ with $$\mathbb{C}\langle T\rangle_ k$$, then $${\mathcal P}\widehat {\otimes}\mathbb{C}$$ is a quotient of $$\mathbb{C}\langle T\rangle_ k$$,and $$\varphi(\psi_ \mathbb{C})\in {\mathcal R} \widehat{\otimes} \mathbb{C}$$ may be regarded as a matrix whose entries are determined by (the images) of power series in $$\mathbb{C}\langle T\rangle_ k$$. Further when $$I=0$$, the semi-simplicity of $$\varphi(\psi_ \mathbb{C})$$ (and hence that of the universal deformations) is discussed briefly.

##### MSC:
 11S20 Galois theory 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 14F30 $$p$$-adic cohomology, crystalline cohomology 20G10 Cohomology theory for linear algebraic groups
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