The analytic variation of p-adic Hodge structure. (English) Zbl 0662.12018

Let K be a p-adic field and E a finite extension of \({\mathbb{Q}}_ p\) contained in K. Let \({\mathfrak O}_ E\) be the ring of integers in E and \({\mathfrak O}_ E[[T]]={\mathfrak O}_ E[[T_ 1,...,T_{\ell}]]\) be the power series ring in \(\ell\) variables over \({\mathfrak O}_ E\), given the usual topology. Denote by I an ideal of \({\mathfrak O}_ E[[T]]\). The main interest of this paper is in continuous representations \(U_ T:\quad {\mathfrak G}\to GL_ n({\mathfrak O}_ E[[T]]/I)\) where \({\mathfrak G}\) is the absolute Galois group of K.
The author introduces compositions of type \(\overline{\alpha}U_ T\) where \(\overline{\alpha}\) is induced by a continuous algebra homomorphism \(\alpha:\quad {\mathfrak O}_ E[[T]]\to {\mathfrak O}_ E\) which vanishes on I. These specializations \(\overline{\alpha}U_ T\) are the representation for which an operator \(\phi\) and a generalized Hodge-Tate decomposition can be defined. The author proves that the characteristic polynomial of \(\phi\) varies analytically as a function of \(\alpha\).
Reviewer: W.Govaerts


11S20 Galois theory
13J05 Power series rings
22E50 Representations of Lie and linear algebraic groups over local fields
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