Hélène Esnault
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.
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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).
Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.
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.
We give a proof that the p-adic multi-zeta values satisfy the Drinfel'd-Ihara relations.
P-adic Gamma classes and overconvergent Frobenius structures for quantum connections
研究单调辛流形小量子联络的超收敛 Frobenius 结构，猜想其常数项由 Morita p-adic Gamma 示性类给出，并在环面 Fano 簇与 Grassmann 簇上得到证明及实验支持。
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 …
Logarithmic geometry and Frobenius, II
本文通过研究对数混合Hodge结构与$\ell$-adic性质的类比，证明了与SL(2)-轨道定理相关的Hodge理论中某些定理的$\ell$-adic类比。
We show that one can express Frobenius transformation on middle-dimensional p-adic cohomology of Calabi-Yau threefold in terms of mirror map and instanton numbers. We express the mirror map in terms of Frobenius transformation on p-adic cohomology . We discuss a $p$-adic interpretation of the conjecture about integrality of Gopakumar-Vafa invariants.
We prove results that, for a certain class of non-compact Calabi-Yau threefolds, relate the Frobenius action on their $p$-adic cohomology to the Frobenius action on the $p$-adic cohomology of the corresponding curves. In the appendix, we describe our interpretation of the Griffiths-Dwork method.
Frobenius structure and $p$-adic zeta values
本文证明了对于$n$维Calabi-Yau超曲面的单纯形和超八面体族，其$p$-adic Frobenius结构的矩阵元中确实出现$p$-adic zeta值，验证了Candelas等人的猜想。
Perfect-Prismatic F-Crystals and p-adic Shtukas in Families
本文构造了 Perfect-Prismatic F-Crystals 与族中 p-adic Shtukas 两范畴间的等价，统一了形式与解析 Frobenius 线性对象，适用于任意 I-完备环及仿射平坦群作用，无需 Frobenius 线性结构。
We produce a canonical filtration for locally free sheaves on an open p-adic annulus equipped with a Frobenius structure. Using this filtration, we deduce a conjecture of Crew on p-adic differential equations, analogous to Grothendieck's local monodromy theorem (also a consequence of results of Andre and of Mebkhout). Namely, given a finite locally free sheaf on an open p-adic annulus with a connection and a compatible Frobenius structure, the corresponding module admits a basis over a finite cover of the annulus on which the connection acts via a nilpotent matrix. Note: this preprint improves…
Higher Residue Pairing for $p$-adic Isocrystals and the $p$-adic Riemann--Hilbert Correspondence
本文提出一种$p$-adic类比的higher residue pairing，通过扭曲的相对de Rham-Witt复形构建，连接晶体几何与残差理论，并为$p$-adic原始形式等理论提供基础。
Analytic Geometry and Hodge-Frobenius Structure
本文研究高维p进解析几何中的Frobenius结构及相关p进泛函分析，依托既有研究纲领与方法探究广义p进Hodge结构形变，为相关上同调、岩泽理论及非交换玉河数猜想研究奠基。
Let $X\subset ¶^n$ be a possibly singular hypersurface of degree $d\le n$, defined over a finite field $k$. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its $\ell$-adic cohomology $H^i(\bar{X}, \Q_\ell)$, for $\ell \neq {\rm char}(k)$, are divisible by $q$, without using the result on the existence of rational points by Ax and Katz.
p-adic iterated integration on semistable curves
本文以幂零对数刚性基本群重构半稳定曲线上的 p-adic 迭代积分，借助 Frobenius 不变量与对偶图阐明 Berkovich–Coleman 积分，并将 Vologodsky 积分解释为沿组合典范路径的积分。
On the variation of the Frobenius in a non abelian Iwasawa tower
本文研究有限域上多自同构代数簇Frobenius算子特征值的ℓ进性质，证明一类岩泽塔中étale上同调Frobenius特征多项式的ℓ进收敛性，给出相关序列收敛速率，并提出多自同构曲线超奇异性的精确判据，推广统一了过往结果。
Two classical rings of invariants are shown to be Frobenius split: for the special linear group acting on the direct sum of several copies of the defining representation and several copies of the dual of the defining representation; and for the special orthogonal group acting on several copies of the defining representation.
Frobenius intertwiners for q-difference equations
本文研究射影空间余切丛上的量子差分方程，证明其具有 Frobenius 作用，给出交织子常数项的显式公式，并推广到 Dwork 的 p-adic 超几何与 Bessel 方程。
p-adic Rigidity and Supercongruences at Rank Two Attractors
本文研究秩二吸引子引发的p-adic刚性，提出上同调判据证明Frobenius不动点与超同余，并应用于Hulek--Verrill A_4族及镜像四次K3铅笔。
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.
To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $\ell$-adic counterpart of the GKZ hypergeometric system, which we call the $\ell$-adic GKZ hypergeometric sheaf. It is an object in the derived category of $\ell$-adic sheaves on the affine space over a finite field. Traces of Frobenius on stalks of this object at rational points of the affine space define the hypergeometric functions over the finite field introduced by Gel…
Revisiting Dwork cohomology: Visibility and divisibility of Frobenius eigenvalues in rigid cohomology
该研究通过Dwork方法探究有限域上代数簇紧支集刚性上同调的Frobenius特征值，得到两个算术推论：仿射簇zeta函数可见证相关特征值，且刚性上同调的q-可除性下界优于ℓ进上同调的已知结果。
We compute a basis for the p-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give p-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.
We introduce a "limiting Frobenius structure" attached to any degeneration of projective varieties over a finite field of characteristic p which satisfies a p-adic lifting assumption. Our limiting Frobenius structure is shown to be effectively computable in an appropriate sense for a degeneration of projective hypersurfaces. We conjecture that the limiting Frobenius structure relates to the rigid cohomology of a semistable limit of the degeneration through an analogue of the Clemens-Schmidt exact sequence. Our construction is illustrated, and conjecture supported, by a selection of explicit ex…
$p$-adic Berglund-Hübsch Duality
本文利用 p-进 D-模方法研究 Berglund-Hübsch 对偶，证明 Frobenius 自同态与该对偶在显式对角算子意义下交换。
In this paper, we show that the Galois representations provided by $\ell$-adic cohomology of proper smooth varieties, and more generally by $\ell$-adic intersection cohomology of proper varieties, over any field, are orthogonal or symplectic according to the degree. We deduce this from a preservation result of orthogonal and symplectic pure perverse sheaves by proper direct image. We show moreover that the subgroup of the Grothendieck group generated by orthogonal pure perverse sheaves of even weights and symplectic pure perverse sheaves of odd weights are preserved by Grothendieck's six opera…
We prove a statement on p-adic continuity of matrices of coefficients of the logarithm of the Artin-Mazur formal group law associated to the middle cohomology of a hypersurface. As Jan Stienstra discovered in 1986, the entries of these matrices are coefficients of powers of the equation of the hypersurface, and in certain cases they satisfy congruences of Atkin and Swinnerton-Dyer type. These congruences imply that eigenvalues of our limiting matrices are eigenvalues of Frobenius of zero p-adic valuation on the middle crystalline cohomology of the fibre at p.
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。
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…
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 $\mathrm{char}(\overline{\mathbb{F}}_q) \neq 2,3$.
We introduce a new category of coefficients for p-adic cohomology called constructible isocrystals. Conjecturally, the category of constructible isocrystals endowed with a Frobenius structure is equivalent to the category of perverse holonomic arithmetic D-modules. We prove here that a constructible isocrystal is completely determined by any of its geometric realizations.
$\ell$-Independence, slopes, and weights of $\ell$-adic cohomology groups of rigid spaces
针对不同素数ℓ和p的局部域上的刚性空间，研究ℓ进上同调群的相关性质，证明Weil群正部作用的迹交错和与ℓ无关且为整数，给出几何Frobenius提升特征值p进赋值的最优界，推广相关结果并证明Lu-Zheng猜想。
In the 1960s, Dwork developed a p-adic cohomology theory of de Rham type for varieties over finite fields, based on a trace formula for the action of a Frobenius operator on certain spaces of p-adic analytic functions. One can consider a purely algebraic analogue of Dwork's theory for varieties over a field of characteristic zero and ask what is the connection between this theory and ordinary de Rham cohomology. N. Katz showed that Dwork cohomology coincides with the primitive part of de Rham cohomology for smooth projective hypersurfaces, but the exact relationship for varieties of higher cod…
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 …
The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the $\ell$-adic cohomology of the arrangement (as a variety). We describe the eigenvalues of the Frobenius map acting on this cohomology, which corresponds to a finer decomposition of the zeta function. The $\ell$-adic cohomology groups and their decomposition into eigenspaces are s…
In the late '60s, B. Dwork studied a Frobenius structure compatible with the classical hypergeometric differential equation with parameters $\left(\frac{1}{2},\frac{1}{2} ; 1 \right)$ by analyzing behavior of solutions of the differential equation under Frobenius transformation. Recently, P. Scholze conjectured the existence of $q$-de Rham cohomology groups for any $\mathbb{Z}$-scheme. In this paper, we give a Frobenius structure compatible with the $q$-hypergeometric differential equation with parameters $(q^{\frac12},q^{\frac12};q)$ by showing a $q$-analogue of some results of Dwork. This co…
We prove that, modulo any power of a prime $p$, the absolute integral closure of an excellent noetherian domain is Cohen-Macaulay. A graded analog is also established, yielding variants of Kodaira vanishing "up to finite covers" in mixed characteristic. Our main tools are (log) prismatic cohomology (which yields a Frobenius action in mixed characteristic) and the $p$-adic Riemann-Hilbert functor for constructible étale $\mathbf{F}_p$-sheaves on varieties over a $p$-adic field (which almost controls perfectified prismatic cohomology).
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。
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。
Arithmetic geometry of quantum connections on Calabi-Yau $3$-folds
研究量子连接在卡拉比-叶三重流形上的算术结构，建立模p量子cohomology上的逆Cartier算子与Frobenius拉回的关系，提出关于更一般半正辛流形上量子连接的猜想。
Actions of nilpotent groups on complex algebraic varieties
本文通过基变换结合有限域计数与 p 进 Lie 群理论，研究幂零群对复代数簇的忠实作用，并给出有限 p-群线性嵌入及维数下界结论。
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…
We show that Aomoto's $q$-deformation of de Rham cohomology arises as a natural cohomology theory for $Λ$-rings. Moreover, Scholze's $(q-1)$-adic completion of $q$-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in mixed characteristic equipped with a suitable lift of Frobenius. If we attach $p$-power roots of $q$, the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.
Rank 2 $\ell$-adic local systems and Higgs bundles over a curve
针对有限域上曲线补集的秩2 ℓ-adic局部系统，研究被Frobenius迭代固定的给定tame局部单模的系统数量，证实Deligne猜想并将结果用Higgs丛数量表示。
In this paper we investigate the arithmetic aspects of the theory of $\mathcal{E}_K^\dagger$-valued rigid cohomology introduced and studied in [11,12]. In particular we show that these cohomology groups have compatible connections and Frobenius structures, and therefore are naturally $(φ,\nabla)$-modules over $\mathcal{E}_K^\dagger$ whenever they are finite dimensional. We also introduce a category of `absolute' coefficients for the theory; the same results are true for cohomology groups with coefficients. We moreover prove a $p$-adic version of the weight monodromy conjecture for smooth (not …
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