The first part of the paper develops the theory of $m$-shifted $π$-typical Witt vectors which can be viewed as subobjects of the usual $π$-typical Witt vectors. We show that the shifted Witt vectors admit a delta structure that satisfy a canonical identity with the delta structure of the usual $π$-typical Witt vectors. Using this theory, we prove that the generalized kernels of arithmetic jet spaces are jet spaces of the kernel at the first level. This also allows us to interpret the arithmetic Picard-Fuchs operator geometrically.
For a $π$-formal group scheme $G$, by a previous construction, one attaches a canonical filtered isocrystal $\mathbf{H}_δ(G)$ associated to the arithmetic jet spaces of $G$. In the second half of our paper, we show that $\mathbf{H}_δ(A)$ is of finite rank if $A$ is an abelian scheme. We also prove a strengthened version of a result of Buium on delta characters on abelian schemes. As an application, for an elliptic curve $A$ defined over $\mathbb{Z}_p$, we show that our canonical filtered isocrystal $\mathbf{H}_δ(A) \otimes \mathbb{Q}_p$ is weakly admissible. In particular, if $A$ does not admit a lift of Frobenius, we show that $\mathbf{H}_δ(A) \otimes \mathbb{Q}_p$ is isomorphic to the first crystalline cohomology $\mathbf{H}^1_{\mathrm{cris}}(A) \otimes \mathbb{Q}_p$ in the category of filtered isocrystals. On the other hand, if $A$ admits a lift of Frobenius, then $\mathbf{H}_δ(A) \otimes \mathbb{Q}_p$ is isomorphic to the sub-isocrystal $H^0(A,Ω_A) \otimes \mathbb{Q}_p$ of $\mathbf{H}^1_{\mathrm{cris}}(A) \otimes \mathbb{Q}_p$.
We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.
In this article, we prove the comparison theorem between the relative log de Rham-Witt cohomology and the relative log crystalline cohomology for a log smooth saturated morphism of fs log schemes satisfying certain condition. Our result covers the case where the base fs log scheme is etale locally log smooth over a scheme with trivial log structure or the case where the base fs log scheme is hollow, and so it generalizes the previously known results of Matsuue. In Appendix, we prove that our relative log de Rham-Witt complex and our comparison map are compatible with those of Hyodo-Kato.
For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic étale cohomology $H^i_{\mathrm{ét}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.
We introduce rigid syntomic cohomology for strictly semistable log schemes over a complete discrete valuation ring of mixed characteristic (0,p). In case a good compactification exists, we compare this cohomology theory to Nekovář-Nizioł's crystalline syntomic cohomology of the generic fibre. The main ingredients are a modification of Große-Klönne's rigid Hyodo-Kato theory and a generalisation of it for strictly semistable log schemes with boundary.
We define an overconvergent version of the Hyodo-Kato complex for semistable varieties $Y$ over perfect fields of positive characteristic, and prove that its hypercohomology tensored with $\mathbb{Q}$ recovers the log-rigid cohomology when $Y$ is quasi-projective. We then describe the monodromy operator using the overconvergent Hyodo-Kato complex. Finally, we show that overconvergent Hyodo-Kato cohomology agrees with log-crystalline cohomology in the projective semistable case.
In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus' prediction for some families of abelian varieties. These families include abelian varieties that have both prime dimension and nontrivial endomorphism ring. The proof is based on …
We survey crystalline cohomology, crystals, and formal group laws with an emphasis on geometry. We apply these concepts to K3 surfaces, and especially to supersingular K3 surfaces. In particular, we discuss stratifications of the moduli space of polarized K3 surfaces in positive characteristic, Ogus' crystalline Torelli theorem for supersingular K3 surfaces, the Tate conjecture, and the unirationality of K3 surfaces.
We prove that, for adic étale motives over $\mathbb{C}_p$, the vector bundles on the Fargues-Fontaine curve arising from their Hyodo-Kato cohomology coincide with their de Rham-Fargues-Fontaine cohomologies, where the latter provides an overconvergent refinement of crystalline vector bundles, albeit constructed on the generic fiber. This equivalence is established in the setting of symmetric monoidal $\infty$-categories and respects the full motivic structure. Furthermore, we enrich both realizations with Galois actions, yielding $G_{\breve{\mathbb{Q}}_{p}}$-equivariant solid quasi-coherent sh…
We compare flat cohomology with crystalline syntomic complexes in two cases: 1) $p$-divisible groups over a separated $\mathbb F_p$-scheme with local finite $p$-bases, 2) semi-abelian schemes over a separated irreducible smooth curve.
In this follow-up paper we show that smooth Hodge-proper stacks over $\mathcal O_K$ are $\mathbb Q_p$-locally acyclic: namely the natural map between étale $\mathbb Q_p$-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the $\mathbb Q_p$-case of general conjectures made in our previous work. As a corollary, we get that if a smooth Artin stack over $K$ has a smooth Hodge-proper model over $\mathcal O_K$, its $\mathbb Q_p$-étale cohomology is a crystalline Galois representation. We then also establish a truncated version of the above results in more gener…
On relative and overconvergent de Rham-Witt cohomology for log schemes
我们构造了相对log de Rham-Witt复形。这是Langer-Zink的相对de Rham-Witt复形到log格式的推广。证明了log de Rham-Witt复形的超上同调与相对对数结晶上同调的比较定理。我们构造了相对严格半稳定对数格式的$p$ -进权谱序列.当基对数概型是对数点时，我们证明了它在Witt环的分数域张量后在$E_2$退化。我们还将Davis-Langer-Zink的过收敛de Rham-Witt复形的定义推广到与完满域上具有简单正规交叉因子的光滑概形相关的对数格式$(X,D)$.最后，我们将它的超上同调与$X \setminus D$的刚性上同调进行了比较。
Locally analytic vectors in the completed cohomology of quaternionic Shimura curves
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
