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\(p\)-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].

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