For a proper, smooth scheme X over a p-adic field K, we show that any proper, flat, semistable OK-model X of X whose logarithmic de Rham cohomology is torsion free determines the same OK -lattice inside H i dR (X/K) and, moreover, that this lattice is functorial in X. For this, we extend the results of Bhatt-Morrow-Scholze on the construction and the analysis of an A inf -valued cohomology theory of p-adic formal, proper, smooth O K -schemes X to the semistable case. The relation of the A inf -cohomology to the p-adic étale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine-Jannsen.

We introduce a logarithmic variant of the notion of δ-rings, which we call δ log -rings, and use it to define a logarithmic version of the prismatic site introduced by Bhatt and Scholze. In particular, this enables us to construct the Breuil-Kisin cohomology in the semistable case. Contents 1. Introduction 1 2. δ log -rings 6 3. Logarithmic prisms 13 4. The logarithmic prismatic site 18 5. The crystalline comparison 22 6. The Hodge-Tate comparison 27 7. Logarithmic q-crystalline cohomology and logarithmic q-de Rham complexes 35 8. Comparison with AΩ 43 Appendix A. Complements on logarithmic geometry 47 References 575 The de Rham comparison can be already proved in some cases by using the crystalline comparison and base change. Say (A, I) = (W (k) [[u]], E(u)), then we take the (φ-twisted) base change to the p-completed PD envelope S of (E(u)) ⊂ W (k)[u], apply the crystalline comparison, and specialize the log crystalline cohomology from S to O K .

We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of $K3$ surfaces over finite fields. We prove that every $K3$ surface of finite height over a finite field admits a characteristic $0$ lifting whose generic fibre is a $K3$ surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a $K3$ surface over a finite field. To obtain these results, we construct an analogue of Kisin’s algebraic group for a $K3$ surface of finite height and construct characteristic $0$ liftings of the $K3$ surface preserving the action of tori in the algebraic group. We obtain these results for $K3$ surfaces over finite fields of any characteristics, including those of characteristic $2$ or $3$ .

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We show that a semisimple overconvergent "absolutely unit-root" F -isocrystal on a geometrically connected smooth variety over a finite field becomes constant over a finite covering.