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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.

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
Full Text: DOI Numdam EuDML
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