## 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

### MSC:

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