We study the unipotent completion $Π^{DR}_{un}(x_0, x_1, X_K)$ of the de Rham fundamental groupoid [De] of a smooth algebraic variety over a local non-archimedean field K of characteristic 0.
We show that the vector space $Π^{DR}_{un}(x_0, x_1, X_K)$ possesses a distinguished element. In the other words, given a vector bundle E on $X_K$ together with a unipotent integrable connection, we have {\sf a canonical} isomorphism $E_{x_0}\simeq E_{x_1}$ between the fibers. The latter construction is a generalization of Colmez's p-adic integration (rk E=2) and Coleman's p-adic iterated integrals ($X_K$ is a curve with good reduction).
In the second part we prove that, if $X_{K_0}$ is a smooth variety over an unramified extension of $\mathbb{Q}_p$ with good reduction and $r \leq \frac{p-1}{2}$ then there is a canonical isomorphism $Π^{DR}_{r}(x_0, x_1, X_{K_0})\otimes B_{DR} \simeq Π^{et}_{r}(x_0, x_1, X_{\overline K_0}) \otimes B_{DR}$ compatible with the action of Galois group (Here $Π^{DR}_{r}(x_0, x_1, X_{K_0})$ is the level r quotient of $Π^{DR}_{un}(x_0, x_1, X_K)$). In particularly, it implies the Crystalline Conjecture for the fundamental group [Shiho] (for $r \leq \frac{p-1}{2}$) .
We introduce the categories of geometric mixed Hodge modules on algebraic varieties over a subfield $k\subset\mathbb C$, and for a prime number $p$, the categories of geometric $p$-adic mixed Hodge modules on algebraic varieties over a subfield $k\subset K\subset\mathbb C_p$ of a p-adic field $K$. We then give an Hodge realization functor on the de derived category of relative motives over $k\subset\mathbb C$, and for a prime number $p$ a $p$-adic Hodge realization functor on the derived category of relative motives over $k\subset K\subset\mathbb C_p$.
We interpret syntomic cohomology of Nekovář-Nizioł as a $p$-adic absolute Hodge cohomology. This is analogous to the interpretation of Deligne-Beilinson cohomology as an absolute Hodge cohomology by Beilinson and generalizes the results of Bannai and Chiarellotto, Ciccioni, Mazzari in the good reduction case, and of Yamada in the semistable reduction case. This interpretation yields a simple construction of the syntomic descent spectral sequence and its degeneration for proper and smooth varieties. We introduce syntomic coefficients and show that in dimension zero they form a full triangulated…
$p$-adic Hodge Theory is one of the most powerful tools in modern Arithmetic Geometry. In this survey, we will review $p$-adic Hodge Theory for algebraic varieties, present current developments in $p$-adic Hodge Theory for analytic varieties, and discuss some of its applications to problems in Number Theory. This is an extended version of a talk at the Jubilee Congress for the 100th anniversary of the Polish Mathematical Society, Kraków, 2019.
The aim of this paper is to give an application of p-adic Hodge theory to stringy Hodge numbers introduced by V. Batyrev for a mathematical formulation of mirror symmetry. Since the stringy Hodge numbers of an algebraic variety are defined by choosing a resolution of singularities, the well-definedness is not clear from the definition. We give a proof of the well-definedness based on arithmetic results such as p-adic integration and p-adic Hodge theory. Note that another proof of the well-definedness was already obtained by V. Batyrev himself by motivic integration. This is a generalization of…
We construct infinitely many non-isotrivial families of abelian varieties over given four punctured projective lines. These families lead to algebraic solutions of Painleve VI equation. Finally, based on a recent paper by Lin-Sheng-Wang, we prove a complete characterization for the locus of motivic Higgs bundles in the moduli space as fixed points of an ``additive'' self-map. This is a note based on the lecture given by the second named author on 04 Nov. 2022 at Tsinghua University.
We present a detailed overview of the construction of the A_inf-cohomology theory from the preprint "Integral p-adic Hodge theory", joint with B. Bhatt and P. Scholze. We focus particularly on the p-adic analogue of the Cartier isomorphism via relative de Rham--Witt complexes.
We revisit the Harder-Narasimhan stratification on a minuscule $p$-adic flag variety, by the theory of modifications of $G$-bundles on the Fargues-Fontaine curve. We compare the Harder-Narasimhan strata with the Newton strata introduced by Caraiani-Scholze. As a consequence, we get further equivalent conditions in terms of $p$-adic Hodge-Tate period domains for fully Hodge-Newton decomposable pairs. Moreover, we generalize these results to arbitrary cocharacters case by considering the associated $B_{dR}^+$-affine Schubert varieties. Applying Hodge-Tate period maps, our constructions give appl…
Given an integral p-adic variety, we observe that if the integral Hodge--de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral p-adic Hodge theory, we show that such a nice behavior is guaranteed if the p-adic variety can be lifted to an analogue of second Witt vectors and satisfies some bound on dimension and ramification index. This is a (ramified) mixed characteristic analogue of results due to Deligne--Illusie and Fontaine--Messing. Lastly, we discuss an example illustrating the necessity of t…
Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. Following the one we initiated previously, we present an overview of a new monograph developing new features of the p-adic Simpson correspondence, inspired by our construction of the relative Hodge-Tate spectral sequence. First, we address the connection to Hodge-Tate local systems. Second, we establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, we expand the scope of our…
Faltings' approach in $p$-adic Hodge theory can be schematically divided into two main steps: firstly, a local reduction of the computation of the $p$-adic étale cohomology of a smooth variety over a $p$-adic local field to a Galois cohomology computation and then, the establishment of a link between the latter and differential forms. These relations are organized through Faltings ringed topos whose definition relies on the choice of an integral model of the variety, and whose good properties depend on the (logarithmic) smoothness of this model. Scholze's generalization for rigid analytic vari…
We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e. for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the pro-étale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the pro-étale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another in…
Sen attached to each p-adic Galois representation of a p-adic field a multiset of numbers called generalized Hodge-Tate weights. In this paper, we discuss a rigidity of these numbers in a geometric family. More precisely, we consider a p-adic local system on a rigid analytic variety over a p-adic field and show that the multiset of generalized Hodge-Tate weights of the local system is constant. The proof uses the p-adic Riemann-Hilbert correspondence by Liu and Zhu, a Sen-Fontaine decompletion theory in the relative setting, and the theory of formal connections. We also discuss basic propertie…
We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-etale site, which makes all constructions completely functorial.
We relate various approaches to coefficient systems in relative integral $p$-adic Hodge theory, working in the geometric context over the ring of integers of a perfectoid field. These include small generalised representations over $A_{\text{inf}}$ inspired by Faltings, modules with q-connection in the sense of q-de Rham cohomology, crystals on the prismatic site of Bhatt--Scholze, and q-deformations of Higgs bundles.
We construct Λ-adic de Rham and crystalline analogues of Hida's ordinary Λ-adic etale cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \Q_p, we prove appropriate finiteness and control theorems in each case. We then employ integral p-adic Hodge theory to prove Λ-adic comparison isomorphisms between our cohomologies and Hida's etale cohomology. As applications of our work, we provide a "cohomological" construction of the family of (ϕ,Γ)-modules attached to Hida's ordinary Λ-adic etale cohomology by Dee, and we give a new and purely…
Given a proper, smooth (formal) scheme over the ring of integers of $\mathbb C_p$, we prove that if the crystalline cohomology of its special fibre is torsion-free then the $p$-adic étale cohomology of its generic fibre is also torsion-free. In this announcement we sketch the proof, which relies on the construction of a new cohomology theory interpolating between crystalline and étale cohomology. Further details and results, including a comparison isomorphism, will be presented in the full forthcoming article.
The Hodge-Tate spectral sequence for a proper smooth variety over a p-adic field provides a framework for us to revisit Faltings' approach to p-adic Hodge theory and to fill in many details. The spectral sequence is obtained from the Cartan-Leray spectral sequence for the canonical projection from the Faltings topos to the étale topos of an integral model of the variety. Its abutment is computed by Faltings' main comparison theorem from which derive all comparison theorems between p-adic étale cohomology and other p-adic cohomologies, and its initial term is related to the sheaf of differentia…
Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum of Hodge type. Let $p$ be an odd prime such that $\mathsf{G}_{\mathbb{Q}_p}$ splits after a tamely ramified extension and $p\nmid |π_1(\mathsf{G}^{\rm der})|$. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and $\mathsf{G}_{\mathbb{Q}_p}$ is either unramified or residually split, we prove the generalisation of Mantovan's formula for the $l$-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction…
This work is devoted to the study of integral $p$-adic Hodge theory in the context of Artin stacks. For a Hodge-proper stack, using the formalism of prismatic cohomology, we establish a version of $p$-adic Hodge theory with the étale cohomology of the Raynaud generic fiber as an input. In particular, we show that the corresponding Galois representation is crystalline and that the associated Breuil-Kisin module is given by the prismatic cohomology. An interesting new feature of the stacky setting is that the natural map between étale cohomology of the algebraic and the Raynaud generic fibers is…
We prove that for any proper smooth formal scheme $\frak X$ over $\mathcal O_K$, where $\mathcal O_K$ is the ring of integers in a complete discretely valued nonarchimedean extension $K$ of $\mathbb Q_p$ with perfect residue field $k$ and ramification degree $e$, the $i$-th Breuil-Kisin cohomology group and its Hodge-Tate specialization admit nice decompositions when $ie<p-1$. Thanks to the comparison theorems in the recent works of Bhatt, Morrow and Scholze, we can then get an integral comparison theorem for formal schemes when the cohomological degree $i$ satisfies $ie<p-1$, which generalize…
We give a new proof of the Tate-Voloch conjecture, in the situation where the ambient variety is a semiabelian variety defined over Qp. Our proof is new in the sense that it avoids any reference to algebraic model theory or p-adic Hodge theory.
The basic admissible locus $\mathcal{F}(G, μ, b)^a$ inside the flag variety $\mathcal{F}(G, μ)$, attached to a reductive group $G$ with a minuscule cocharacter $μ$ of $G$, is a $p$-adic analogue of the complex analytic period spaces. It has an algebraic approximation $\mathcal{F}(G, μ, b)^{wa}$ inside the flag variety, called the weakly admissible locus. On the flag variety $\mathcal{F}(G, μ)$, we have the Newton stratification which has the admissible locus as its unique open stratum. In this paper, we study the relation between the Newton strata and the weakly admissible locus. We show that …
We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of C_p. It takes values in a mixed-characteristic analogue of Dieudonne modules, which was previously defined by Fargues as a version of Breuil-Kisin modules. Notably, this cohomology theory specializes to all other known p-adic cohomology theories, such as crystalline, de Rham and etale cohomology, which allows us to prove strong integral comparison theorems. The construction of the cohomology theory relies on Faltings's almost purity theorem, along with a certain functor $Lη$ on the derived cate…
We study the p-adic deformation properties of algebraic cycle classes modulo rational equivalence. We show that the crystalline Chern character of a vector bundle lies in a certain part of the Hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal pro-class in K-theory on the p-adic scheme.
For a number field, we have a Tannaka category of mixed Tate motives at our disposal. We construct p-adic points of the associated Tannaka group by using p-adic Hodge theory. Extensions of two Tate objects yield functions on the Tannaka group, and we show that evaluation at our p-adic points is essentially given by the inverse of the Bloch-Kato exponential map.
Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation
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…
For an algebraic variety $X$ of dimension $d$ with totally degenerate reduction over a $p$-adic field (definition recalled below) and an integer $i$ with $1\leq i\leq d$, we define a rigid analytic torus $J^i(X)$ together with an Abel-Jacobi mapping to it from the Chow group $CH^i(X)_{hom}$ of codimension $i$ algebraic cycles that are homologically equivalent to zero modulo rational equivalence. These tori are analogous to those defined by Griffiths using Hodge theory over the complex numbers. We compare and contrast the complex and $p$-adic theories. Finally, we examine a special case of a $p…
In a previous paper, we constructed a category of (phi, Gamma)-modules associated to any adic space over Q_p with the property that the etale (phi, Gamma)-modules correspond to etale Q_p-local systems; these involve sheaves of period rings for Scholze's pro-etale topology. In this paper, we first extend Kiehl's theory of coherent sheaves on rigid analytic spaces to a theory of pseudocoherent sheaves on adic spaces, then construct a corresponding theory of pseudocoherent (phi, Gamma)-modules. We then relate these objects to a more explicit construction in case the space comes equipped with a su…
Motivated by a question of Hansen and Li, we show that a smooth and proper rigid analytic space $X$ with projective reduction satisfies Hodge symmetry in the following situations: (1) the base non-archimedean field $K$ is of residue characteristic zero, (2) $K$ is $p$-adic and $X$ has good ordinary reduction, (3) $K$ is $p$-adic and $X$ has "combinatorial reduction."' We also reprove a version of their result, Hodge symmetry for $H^1$, without the use of moduli spaces of semistable sheaves. All of this relies on cases of Kato's log hard Lefschetz conjecture, which we prove for $H^1$ and for lo…
In this paper, we are going to establish a simultaneous generalization of the relative Iwasawa theory proposed by Kedlaya-Pottharst and the relative $p$-adic Hodge theory after Kedlaya-Liu. We call this Hodge-Iwasawa theory in the sense that one could apply the theory to study noncommutative Iwasawa cohomology and noncommutative Iwasawa theories in families and meanwhile one could apply the theory to study the deformation theory of étale local systems or families of representations of fundamental groups or the equivariant constructible $p$-adic sheaves, with more sophisticated point of view co…
The $C$-function of $T$-adic exponential sums is studeid. An explicit arithmetic bound is established for the Newton polygon of the $C$-function. This polygon lies above the Hodge polygon. It gives a sup-Hodge bound of the $C$-function of $p$-power order exponential sums.
This is a resume of the author's talk at the Worhshop on Arithmetic, Geometry and Physics around Calabi-Yau Varieties and Mirror Symmetry (July 23-29, 2001), the Fields Institute. The aim of this note is to prove that birational smooth minimal models over C have equal Hodge numbers in all dimensions by an arithmetic method. Our method is a refinement of the method of V. Batyrev and C.-L. Wang on Betti numbers who used p-adic integration and the Weil conjecture. Our ingredient is to use further arithmetic results such as the Chebotarev density theorem and p-adic Hodge theory.
The purpose of this article is to prove a "Newton over Hodge" result for finite characters on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $ρ:π_1^{et}(V) \to \mathbb{C}^\times$. In this article, we prove a lower bound on the Newton polygon of the $L$-function $L(ρ,s)$. The estimate depends on monodromy invariants of $ρ$: the Swan conductor and the local exponents. Under certain nondegeneracy assumptions this lower bound agrees with the irregular Hodge filt…
