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Semi-stable conjecture of Fontaine-Jannsen: A survey. (English) Zbl 1041.14003
Berthelot, Pierre (ed.) et al., $$p$$-adic cohomology and arithmetic applications (II). Paris: Société Mathématique de France (ISBN 2-85629-117-1/pbk). Astérisque 279, 323-370 (2002).
From the paper: In these notes, we give an outline of the proof of O. Hyodo and K. Kato [in: Périodes $$p$$-adiques, Sémin. Bures-sur-Yvette 1988, Exposé V, Astérisque 223, 221–268 (1994; Zbl 0852.14004)], K. Kato ibid., Exposé VI, Astérisque 223, 269–293 (1994; Zbl 0847.14009) and T. Tsuji [Invent. Math. 137, 233–411 (1999; Zbl 0945.14008)] of the conjecture of J.-M. Fontaine and U. Jannsen [in: Galois groups over $$\mathbb{Q}$$, Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, 315–359 (1989; Zbl 0703.14010)] and J.-M. Fontaine [in: Périodes $$p$$-adiques. Sém. Bures-sur-Yvette 1988, Exposé III, Astérisque 223, 113–184 (1994; Zbl 0865.14009; §6] on the $$p$$-adic étale cohomology of a proper smooth variety over a $$p$$-adic field with semi-stable reduction. This conjecture compares the two $$p$$-adic cohomologies: $$p$$-adic étale cohomology and de Rham cohomology associated to a proper smooth variety over a $$p$$-adic field with semi-stable reduction; it especially asserts that these two cohomologies with their additional structures can be reconstructed from each other. Our proof uses syntomic cohomology, which was introduced by J.-M. Fontaine and W. Messing, as a bridge between the two cohomologies.
In the appendix, we also show that the semi-stable conjecture implies the de Rham conjecture thanks to the alteration of de Jong.
For the entire collection see [Zbl 0990.00020].

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F40 de Rham cohomology and algebraic geometry