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The torsor between \(p\)-adic étale cohomology and crystalline cohomology; the abelian case. (Torseur entre cohomologie étale \(p\)-adique et cohomologie cristalline; le cas abélien.) (French) Zbl 0746.14007

The author studies the functor \(V\to D(V)\) where (a) \(K\) is a local field of characteristic 0 with perfect residual field \(k\) of characteristic \(p\), (b) \(V\) is an abelian \(p\)-adic crystalline representation of the Galois group \(G_ K\) of the algebraic closure \(\overline K\) of \(K\), and (c) \(D(V)\) is a filtered Dieudonné module. Note that \(D(V)=(V\otimes_{\mathbb{Q}_ p}B_{cris})^{G_ K}\) where \(B_{cris}\) is a \(P[G_ K]\)-algebra introduced by Fontaine and, if \({\mathfrak O}_ K\) is the ring of integers of \(K\) and \(W\) is the ring of Witt vectors contained in \({\mathfrak O}_ K\), then \(P\) is the quotient field of \(W\). Omitting extensive detail, the first main result of the author shows that: \[ i\circ F\circ N_{N/P}(h_ K(\pi_ K))=f\circ i \] where (a) \(i\) is a point of the torsor \(I_ V\), (b) \(F\) is the frobenius of \(D(V)\), (c) \(N_{K/P}\) is the norm, (d) \(h_ K(\pi_ K)\) is an action on \(D(V)\) defined by means of certain torus \(T_ E\) on \(\mathbb{Q}_ p\) (where \(T_ E(A)=(E\otimes_{\mathbb{Q}_ p}A)^*\) with \(E\) a finite extension of \(\mathbb{Q}_ p\) contained in \(K)\), and (e) \(f\) is the \(\sigma\)- linear endomorphism \(id\otimes\sigma\) of \(V\otimes_{\mathbb{Q}_ p}P\), \(\sigma\) being the frobenius of \(k\).
A similar theorem then extends the preceding result to motifs of type \(CM\) on \(\overline\mathbb{Q}\).

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14L05 Formal groups, \(p\)-divisible groups
11S20 Galois theory
14M17 Homogeneous spaces and generalizations
14K05 Algebraic theory of abelian varieties
13K05 Witt vectors and related rings (MSC2000)
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References:

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