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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