Étale cohomology of rank one $\ell$-adic local systems in positive characteristic

p-adic cohomology

This is a survey of some recent developments concerning the p-adic cohomology of algebraic varieties over fields of positive characteristic and local fields of mixed characteristic, plus some related areas like p-adic Hodge theory.

Computing zeta functions via p-adic cohomology

We survey some recent applications of p-adic cohomology to machine computation of zeta functions of algebraic varieties over finite fields of small characteristic, and suggest some new avenues for further exploration.

On Beilinson's equivalence for $p$-adic cohomology

In this short note, we show a p-adic analogue of Beilinson's equivalence comparing two derived categories: the derived category of holonomic modules and derived category of modules whose cohomologies are holonomic.

Weil-étale Cohomology over $p$-adic Fields

We establish duality results for the cohomology of the Weil group of a $p$-adic field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for discrete Weil modules, which implies Tate-Nakayama Duality. We define Weil-smooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of $\G_m$ on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves.

p-adic derived de Rham cohomology

This paper studies the derived de Rham cohomology of F_p and p-adic schemes, and is inspired by Beilinson's recent work. Generalising work of Illusie, we construct a natural isomorphism between derived de Rham cohomology and crystalline cohomology for lci maps of such schemes, as well logarithmic variants. These comparisons give derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory. Placing these ideas in the skeleton of Beilinson's construction leads to a new proof of Fontaine's crystalline conjecture and Fontaine-Jannsen's semistable conjecture.

p进局部域上刚性空间的ℓ进上同调的斜率与权重

Slopes and weights of $\ell$-adic cohomology of rigid spaces

研究p进局部域上刚性空间的ℓ进上同调，证明其弗罗贝尼乌斯特征值是代数整数并给出p进赋值界，推导权重界证明相关猜想，还给出特定单值纯反常层例子回答问题。

Lectures on Applied $\ell$-adic Cohomology

We describe how a systematic use the deep methods from $\ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz, Laumon allow to make progress on various classical questions from analytic number theory. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 Arizona Winter School.

关于非阿基米德域上正规仿射阿迪空间的整结构层上同调中的挠率

On the torsion in the cohomology of the integral structure sheaf of affinoid adic spaces

本文证明了特征零非阿基米德域上正规仿射阿迪空间的整结构层上同调一致挠，源于Bartenwerfer的注记，部分回答了Hansen和Kedlaya的问题及《非阿基米德苏格兰书》相关问题。

Theory of weights in p-adic cohomology

Let k be a finite field of characteristic p>0. We construct a theory of weights for overholonomic complexes of arithmetic D-modules with Frobenius structure on varieties over k. The notion of weight behave like Deligne's one in the l-adic framework: first, the six operations preserve weights, and secondly, the intermediate extension of an immersion preserves pure complexes and weights.

Local acyclicity in $p$-adic cohomology

We prove an analogue for $p$-adic coefficients of the Deligne--Laumon theorem on local acyclicity for curves. That is, for an overconvergent $F$-isocrystal $E$ on a relative curve $f:U\rightarrow S$ admitting a good compactification, we show that the cohomology sheaves of $\mathbf{R}f_!E$ are overconvergent isocrystals if and only if $E$ has constant Swan conductor at infinity.

On the $p$-adic cohomology of some $p$-adically uniformized varieties

Let $K$ be a finite extension of ${\mathbb Q}_p$ and let $X$ be Drinfel'd's symmetric space of dimension $d$ over $K$. Let $Γ\subset {\rm SL}_{d+1}(K)$ be a cocompact discrete (torsionfree) subgroup and let ${X}_Γ=Γ\backslash {X}$, a smooth projective ${K}$-variety. In this paper we investigate the de Rham and log crystalline (log convergent) cohomology of local systems on $X_Γ$ arising from $K[Γ]$-modules. (I) We prove the monodromy weight conjecture in this context. To do so we work out, for a general strictly semistable proper scheme of pure relative dimension $d$ over a cdvr of mixed chara…

Cohomology of $p$-adic Stein spaces

We compute the $p$-adic étale and the pro-étale cohomologies of the Drinfeld half-space of any dimension. The main input is a new comparison theorem for the $p$-adic pro-étale cohomology of $p$-adic Stein spaces.

Cohomology of Siegel Varieties with p-adic integral coefficients and Applications

Under the assumption that Galois representations associated to Siegel modular forms exist (it is known only for genus at most 2), we show that the cohomology with p-adic integral coefficients of Siegel Varieties, when localized at a non-Eisenstein maximal ideal of the Hecke algebra, is torsion-free, provided the prime p is large with the respect to the weight of the coefficient system. The proof uses p-adic Hodge theory, the dual BGG complex modulo p in order to compute the Hodge-Tate weights for the mod p cohomology. We apply this result to the construction of Hida p-adic families for symplec…

Frobenius action on l-adic Chen-Ruan cohomology

We extend the definition of the Chen-Ruan cohomology ring to smooth, proper, tame, Deligne-Mumford stacks over fields of positive characteristic and prove that a modified version of the Frobenius action preserves the product.

解析空间的算术$p$-进前平展上同调的对偶性

Duality for Arithmetic $p$-adic Pro-étale Cohomology of Analytic Spaces

本研究验证了Colmez等人关于$p$-进数有限扩张上光滑部分紧合空间的算术$p$-进前平展上同调满足对偶性的猜想，借助Fontaine伽罗瓦下降技术由Fargues-Fontaine曲线的几何对偶性推导得出。

关于$p$-进解析空间的上同调，I：基本比较定理

On the cohomology of $p$-adic analytic spaces, I: The basic comparison theorem

本文对$p$-进域代数闭包上的光滑刚性解析簇和dagger簇证明了基本$p$-进比较定理，将稳定范围内的pro-étale上同调表为de Rham上同调的滤过Frobenius特征空间，并通过完美oid空间范畴上的层实现了几何化。

解析簇的紧支集 $p$-adic pro-étale 上同调

Compactly supported $p$-adic pro-étale cohomology of analytic varieties

研究光滑部分真刚性解析簇的紧支集 $p$-adic pro-étale 上同调，证明其与紧支集 syntomic 上同调在稳定范围内的比较定理，并推导出基本图的有限版本。

Integral $p$-adic Hodge theory - announcement

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.

On p-adic comparison theorems for rigid analytic varieties, I

We compute, in a stable range, the arithmetic p-adic etale cohomology of smooth rigid analytic and dagger varieties (without any assumption on the existence of a nice integral model) in terms of differential forms using syntomic methods. The main technical input is a construction of a Hyodo-Kato cohomology and a Hyodo-Kato isomorphism with de Rham cohomology.

解析曲线的$p$-进平展上同调的算术对偶性

Arithmetic duality for $p$-adic pro-étale cohomology of analytic curves

该研究证明了$\textbf{Q}_p$有限扩张上光滑dagger曲线的算术$p$-进pro-étale上同调的庞加莱对偶性，推导结合了谱序列、几何比较定理与经典对偶，还验证了乘积相容性并得到相关上同调的泛函分析性质。

Note on linear relations in Galois cohomology and {é}tale $K$-theory of curves

In this paper we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In \cite{bk13} G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for {é}tale $K$-theory of a curve . This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases this result is the best possible i.e if this condition does not hold we obtain counterexamples. We also give some examples of curves…

一个半稳定概形的log-动机上同调及其p进变形理论

A log-motivic cohomology for semistable varieties and its $p$-adic deformation theory

本文提出了一种半稳定概形的log-动机上同调理论，并探讨了其p进变形理论，通过研究Hyodo-Kato类与Hodge滤波的关系，推广了Bloch-Esnault-Kerz的结论。

开簇与奇异簇的整 p 进上同调理论

Integral p-adic cohomology theories for open and singular varieties

本文研究正特征下开簇与奇异簇的整 p 进上同调，分别借助 cdp/cdh 拓扑及分裂正常一般 étale 超覆盖，给出次数 1 的良好理论并说明其不能推广到高次。

Zelevinsky involution and l-adic cohomology of the Rapoport-Zink tower

In this paper, we investigate how the Zelevinsky involution appears in the l-adic cohomology of the Rapoport-Zink tower. We generalize the result of Fargues on the Drinfeld tower to the Rapoport-Zink towers for symplectic similitude groups.

Divisibility of Frobenius eigenvalues on $\ell$-adic cohomology

v2: For a projective variety defined over a finite field with $q$ elements, it is shown that as algebraic integers, the eigenvalues of the geometric Frobenius acting on $\ell$-adic cohomology have higher than known $q$-divisibility beyond the middle dimension. This sharpens both Deligne's integrality theorem and the cohomological divisibility theorem proven by the first author and N. Katz. Similar lower bounds are proved for the Hodge level for a complex variety beyond the middle dimension, improving earlier results in this direction. We discuss the affine case. The previous version contained …

Integral p-adic Hodge theory

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…

Lefschetz trace formula and l-adic cohomology of Rapoport-Zink tower for GSp(4)

We investigate the alternating sum of the l-adic cohomology of the Rapoport-Zink tower for GSp(4) by the Lefschetz trace formula. Under some assumptions on L-packets of GSp(4) and its inner form, we observe that the local Jacquet-Langlands correspondence appears in the cohomology.

Prisms and Prismatic Cohomology

We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site -- the prismatic site -- to a $p$-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral $p$-adic cohomology theories. As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of $q$-de Rham cohomology as conjectured by the second author, and settle a vanishin…

Cuspidal representations in the l-adic cohomology of the Rapoport-Zink space for GSp(4)

In this paper, we study the l-adic cohomology of the Rapoport-Zink tower for GSp(4). We prove that the smooth representation of GSp_4(Q_p) obtained as the i-th compactly supported l-adic cohomology of the Rapoport-Zink tower has no quasi-cuspidal subquotient unless i=2,3,4. Our proof is purely local and does not require global automorphic methods.

The Geometry of Hida Families and Λ-adic Hodge Theory

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…

$l$-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients

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…

On the action of the Weil group on the l-adic cohomology of rigid spaces over local fields

We investigate the action of the Weil group on the compactly supported l-adic cohomology groups of rigid spaces over local fields. We prove that every eigenvalue of the action is a Weil number when either a rigid space is smooth or the characteristic of the base field is equal to 0. Since a smooth rigid space is locally isomorphic to the Raynaud generic fiber of an algebraizable formal scheme, we can use the techniques of alterations and monodromy spectral sequences in the smooth case. In the general case, we use the continuity theorem of Huber, which requires the restriction on the characteri…

刚性解析簇的有理$p$-进霍奇理论

Rational $p$-adic Hodge theory for rigid-analytic varieties

本文建立$\boldsymbol{\rm C}_p$上无紧性/光滑性假设的刚性解析簇有理$p$-进霍奇理论，通过凝聚与固体形式体系证明其与多类$p$-进上同调的比较关系，验证Le Bras猜想并推广Colmez-Niziol的结果。

$p$-进 étale 上同调中的挠元：注记与猜想

Torsion in $p$-adic étale cohomology: remarks and conjectures

本文研究了非固有光滑形式概形的整 $p$-进 étale 上同调挠元问题，发现仿射情形下挠元可表示为特殊纤维的函子且挠子群有限，并提出了预测更广泛刚性解析簇挠元的猜想。

$p$-进霍奇理论的叠方法

A stacky approach to $p$-adic Hodge theory

本文利用叠方法推广$p$-进霍奇理论中的比较定理，建立带系数的多种上同调比较结果，发展衍射霍奇上同调的叠方法，为波恩大学硕士论文成果。

Bounding Picard numbers of surfaces using p-adic cohomology

Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing the Frobenius action on p-adic cohomology to a small degree of p-adic accuracy. We have implemented this procedure in Magma; using this implementation, we exhibit several examples, such as smooth quartics over F_2 and F_3 with arithmetic Picard number 1, and a smooth quintic over F_2 with geometric …

$p$-adic superspaces and Frobenius

The notion of a $p$-adic superspace is introduced and used to give a transparent construction of the Frobenius map on $p$-adic cohomology of a smooth projective variety over $\zp$ (the ring of $p$-adic integers), as well as an alternative construction of the crystalline cohomology of a smooth projective variety over $\fp$ (finite field with $p$ elements).

$p$-adic 几何 pro-étale 上同调的對偶性

Duality for $p$-adic geometric pro-étale cohomology

本文证明了 $p$-adic 域上光滑部分真刚性解析簇的 $p$-adic 几何 pro-étale 上同调满足 Poincaré 对偶，通过 Fargues-Fontaine 曲线上的固体层及比较定理归结为经典 Serre 对偶。

On the p-adic cohomology of the Lubin-Tate tower

We prove a finiteness result for the p-adic cohomology of the Lubin-Tate tower. For any n>=1 and p-adic field F, this provides a canonical functor from admissible p-adic representations of GL_n(F) towards admissible p-adic representations of Gal_F x D^*, where Gal_F is the absolute Galois group of F, and D/F is the central division algebra of invariant 1/n. Moreover, we verify a local-global-compatibility statement for this correspondence, and compatibility with the patching construction of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin.

p-adic Cohomology and classicality of overconvergent Hilbert modular forms

Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke …

Betti number estimates in p-adic cohomology

Within the framework of Berthelot's theory of arithmetic D-modules, we prove the p-adic analogue of Betti number estimates and we give some standard applications.

Explicit crystalline lattices in rigid cohomology

Motivated by applications in point counting algorithms using p-adic cohomology, we give an explicit description of integral lattices in rigid cohomology spaces that p-adically approximate logarithmic crystalline cohomology modules. These lattices are expressed in terms of the global sections of the twisted logarithmic de Rham complex. We prove the main theorem for smooth proper hypersurfaces with a smooth hyperplane section, then deduce the result for the quotient of such a pair in weighted projective space. We show how these results may be used to reduce the necessary p-adic precision with wh…

Eigenvalues of Frobenius acting on the $\ell$-adic cohomology of complete intersections of low degree

The eigenvalues of Frobenius acting on $\ell$-adic cohomology of a complete intersection over a finite field have the divisibility predicted by the theorem of Ax and Katz. We have corrected some unforgivable typos.

$p$-进数域上 $1$-motives 的凝聚 Weil-étale 实现的对偶性

Duality for the condensed Weil-étale realisation of $1$-motives over $p$-adic fields

本文以凝聚 Weil 群上同调替代 Galois 上同调，引入 $1$-motives 的凝聚 Weil-étale 实现，从而将 Tate 对偶推广到 $p$-进数域上的 $1$-motives。

关于Drinfeld对称空间的$p$-进前平展上同调

On the $p$-adic pro-étale cohomology of Drinfeld symmetric spaces

本研究借助$p$-进霍奇理论相对基本正合序列与凝聚数学新工具，确定了$p$-进域上Drinfeld对称空间的几何$p$-进前平展上同调，给出Colmez-Dospinescu-Niziol定理的新证明，并描述了带系数的正德拉姆周期层的几何前平展上同调。

连续 ℓ-adic 上同调中循环类映射的非单射性

Non-injectivity of the cycle class map in continuous $\ell$-adic cohomology

本文通过构造若干反例，否定回答了 Jannsen 关于连续 ℓ-adic 上同调中整系数循环类映射单射性的问题，并揭示了其对 Chow 群 coniveau 滤过和 Schreieder 超越 Abel-Jacobi 映射的影响。

刚性解析空间的ℓ进平展上同调与邻近环的算术性质

Arithmetic Properties Of $\ell$-adic Étale Cohomology and Nearby Cycles of Rigid-Analytic Spaces

本文研究p进非阿基米德局部域上刚性解析簇的ℓ进平展上同调，建立Frobenius特征值界、强局部单值化定理、邻近环混合性，并用perfectoid分析证明Deligne权-单值化猜想的局部版本，同时发展Deligne拓扑斯上的可构造ℓ进复形理论。

On uniqueness of $p$-adic period morphisms, II

We prove equality of the various $p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the $K$-theoretical uniqueness criterium we had found for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost étale, and motivic constructions. We continue with showing that the $h$-cohomology period morphism agrees with the syntomic and…

Recovering function fields from their integral $\ell$-adic cohomology with the Galois action

In this note, we consider function fields of higher-dimensional algebraic varieties defined over non-local fields, and show how the Galois action on the cohomology such function fields can be used to parameterize their divisorial valuations. By applying a recent theorem of Pop, we observe that, in dimension $\geq 3$, this information is enough to completely determine the function field and the base-field in question.

On $p$-adic absolute Hodge cohomology and syntomic coefficients, I

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-进域上超凯勒簇的算术单值性

Arithmetic monodromy of hyper-Kähler varieties over $p$-adic fields

本文研究p-进域上超凯勒簇的p-进与ℓ-进单值算子，提出并验证了Nagai猜想在算术情形的类比，并引入基于Sen理论的新方法分析p-进上同调。

一个motivic积分$p$-adic上同调

A motivic integral $p$-adic cohomology

本文构造了一个积分$p$-adic上同调，与刚性上同调相容，并在奇点消解条件下证明其与Ertl-Shiho-Sprang的“好”上同调一致，从而得到motivic性质和Künneth公式。

$\ell$-adic étale cohomology of the moduli of stable elliptic fibrations

We determine the $\ell$-adic étale cohomology and the eigenvalues of the geometric Frobenius for the moduli stack $\mathcal{L}_{1,12n} := \mathrm{Hom}_{n}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of stable elliptic fibrations over $\mathbb{P}^{1}$ with $12n$ nodal singular fibers and a marked Weierstrass section over $\overline{\mathbb{F}}_q$ with $\mathrm{char}(\overline{\mathbb{F}}_q) \neq 2,3$.

Zeta functions of nondegenerate hypersurfaces in toric varieties via controlled reduction in $p$-adic cohomology

We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether-Lefschetz divisor in the sense of Hassett.