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A \(p\)-adic property of Hodge classes on abelian varieties. (English) Zbl 0821.14028
Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 2, 293-308 (1994).
Deligne introduced the notion of absolute Hodge classes and proved that on an abelian variety every Hodge class is an absolute Hodge class. In the paper under review the author introduces an analogous notion: an absolute Hodge class \(\gamma_ B\) is De Rham if for every prime \(p\) and every embedding \(\sigma_ p : \overline \mathbb{Q} \to \overline \mathbb{Q}_ p\) we have the compatibility \(I_{DR} (\sigma_ p \gamma_ p) = \sigma_ p \gamma_{DR}\), where \(\gamma_ p\) (resp. \(\gamma_{DR})\) are the images of \(\gamma_ B\) in \(p\)-adic étale (resp. De Rham) cohomology and where \(I_{DR}\) is the comparison map provided by Faltings (in proving the Fontaine conjecture). The author now shows that on an abelian variety over \(\overline \mathbb{Q}\) every Hodge class is De Rham. The proof follows the lines of the proof by Deligne, but replaces his use of the Gauss-Manin connection by some results on the cohomology of families of varieties.
For the entire collection see [Zbl 0788.00054].

14K15 Arithmetic ground fields for abelian varieties
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F30 \(p\)-adic cohomology, crystalline cohomology