## $$p$$-adic periods: A survey.(English)Zbl 0836.14010

Ramanan, S. (ed.) et al., Proceedings of the Indo-French conference on geometry held in Bombay, India, 1989. Delhi: Hindustan Book Agency. 57-93 (1993).
The classical period map is defined for a complete smooth variety $$X$$ over $$\mathbb{C}$$, and it gives an isomorphism $$H^*_{\text{DR}} (X/ \mathbb{C}) @>\approx>> H^* (X (\mathbb{C})$$, $$\mathbb{Q}) \otimes \mathbb{C}$$. The $$p$$- adic period map is defined for a complete smooth variety $$X$$ over the field of fractions $$K$$ of a complete discrete valuation with perfect residue field of characteristic $$p$$, and it relates the de Rham cohomology of $$X$$ (together with its additional structure) to the $$p$$- adic étale cohomology of $$X$$ (together with the action of $$\text{Gal} (K^{\text{al}}/K))$$; it is assumed that $$K$$ has characteristic zero. There is a sequence of successively more precise conjectures, due to the author, that describe what should be true concerning the $$p$$-adic period map for a general $$X$$, for an $$X$$ with semistable reduction, and for an $$X$$ with good reduction. These conjectures have now largely been proved, thanks to the effort of Bloch, Faltings, Fontaine, Hyodo, Kato, Messing, and others.
This article gives a description of the conjectures and results. Except that it is more detailed, and does not discuss proofs, it covers much the same ground as Illusie’s Bourbaki talk [L. Illusie in: Sémin. Bourbaki, Vol. 1989/90, Astérisque 189-190, Exp. No. 726, 325-374 (1990; Zbl 0736.14005)].
For the entire collection see [Zbl 0830.00028].

### MSC:

 14F30 $$p$$-adic cohomology, crystalline cohomology 14F40 de Rham cohomology and algebraic geometry 32G20 Period matrices, variation of Hodge structure; degenerations 11G25 Varieties over finite and local fields 14G20 Local ground fields in algebraic geometry

Zbl 0736.14005