Sascha Orlik
In this paper we consider period domains over local fields for quasi-split reductive groups. We compute the continuous l-adic cohomology with compact support of them in the case of a basic isocrystal. This paper is a continuation of [O2] where we considered the etale cohomology with torsion coefficients of these spaces.
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We construct p-adic period map using derived de Rham cohomology of Illusie and give a simple proof of Fontaine's C_{dR} conjecture.
Prismatic cohomology of rigid analytic spaces over de Rham period ring
本文提出在Fontaine de Rham周期环上为具有l.c.i奇点的刚性解析空间定义prismatic上同调，扩展了相关上同调理论。
Period sheaves via derived de Rham cohomology
本文通过导出de Rham复形，为Scholze的de Rham周期层和Tan-Tong的晶体周期层提供了新的解释。
Galois and Pro-étale Cohomology of Overconvergent de Rham Period Rings
受 p 进微分方程和 p 进几何表示论启发，引入 Fontaine 经典周期环的超收敛变体，计算其 Galois 上同调及相关周期层和周期结构层的上同调。
Period-index in top cohomology over semiglobal fields
本文研究半整体域顶上同调类的周期与指数关系，证明公共槽引理及二者对一般类相等，并给出二次型应用与开放问题。
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…
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。
Generalizing a result of \cite{Z1991, CPZ} about elliptic modular forms, we give a closed formula for the sum of all Hilbert Hecke eigenforms over a totally real number field with strict class number $1$, multiplied by their period polynomials, as a single product of the Kronecker series.
This work gives a version of the Eichler-Shimura isomorphism with a non-abelian $H^1$ in group cohomology. Manin has given a map from vectors of cusp forms to a noncommutative cohomology set by means of iterated integrals. We show Manin's map is injective but far from surjective. By extending Manin's map we are able to construct a bijective map and remarkably this establishes the existence of a non-abelian version of the Eichler-Shimura map.
We show that the continuous cohomology groups of a $ p $-adic reductive group with coefficients in the locally analytic vectors of an admissible $ \mathbb{Q}_p $-Banach space representation are homeomorphic to those with coefficients in the Banach space representation itself. Moreover, we deduce that the canonical topologies on those continuous cohomology groups are Hausdorff and are the uniquely determined finest locally convex topologies.
We compute the p-torsion and p-adic etale cohomologies with compact support of period domains over local fields in the case of basic isocrystals for quasi-split reductive groups. For the p-torsion case, we follow the method used by Orlik in his computations of the l-torsion etale cohomology using as a key new ingredient the computation of Ext groups between mod p generalized Steinberg representations of p-adic groups. For the p-adic case, we don't use Huber's definition of etale cohomology with compact support as Orlik did since it seems to give spaces that are much too big; instead we use con…
We prove that the continuous group cohomology groups of a locally profinite group $ G $ with coefficients in a smooth $ k $-representation $ π$ of $ G $ are isomorphic to the $ \mathrm{Ext}$-groups $ \mathrm{Ext}^i_G(\mathbb{1},π) $ computed in the category of smooth $ k $-representations of $ G $. We apply this to show that if $ π$ is a supersingular $ \overline{\mathbb{F}}_p $-representation of $ \mathrm{GL}_2(\mathbb{Q}_p) $, then the continuous group cohomology of $ \mathrm{SL}_2(\mathbb{Q}_p)$ with values in $ π$ vanishes. Furthermore, we prove that the continuous group cohomology groups …
Squares of toric period integrals in higher cohomology
本文在高次上同调中研究四元数自守表示对应的环面积分周期，将此前对帽积绝对值的计算推广到计算帽积的平方，从而得到与Rankin-Selberg L-函数特殊值相关的Waldspurger型公式。
The goal of this paper is to show that the cohomology of compact unitary Shimura varieties is concentrated in the middle degree and torsion-free, after localizing at a maximal ideal of the Hecke algebra satisfying a suitable genericity assumption. Along the way, we establish various foundational results on the geometry of the Hodge-Tate period map. In particular, we compare the fibres of the Hodge-Tate period map with Igusa varieties.
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. Th…
Patching for étale algebras and the period-index problem for higher degree Galois cohomology groups over Hensel semi-global fields
本文将Harbater-Hartmann的域拼接技术推广到Hensel半全局域，证明étale代数的拼接成立，建立高阶Galois上同调的局部-整体原理，并将周期-指数一致界推广到非素数阶系数群。
We show that the periods of the holomorphic Eisenstein series of level 1, parallel weight 2, on a Hilbert modular surface are not rational, even up to scaling. This is deduced from a study of the mixed Hodge structure on the cohomology of the Hilbert modular surface, where we find extension classes related to the units of the real quadratic field. We prove similar results for Hilbert modular varieties of all dimensions, and produce extensions of Galois representations in étale cohomology. The key point is to study the restriction of the canonical extension of the Hodge bundle to the boundary o…
Classical periods of Eisenstein series and Bernoulli polynomials in the equivariant cohomology of a torus
本文通过取值于currents的群上链为环面的$\ ext{GL}_2$-等变多重对数类提供显式代表元，并基于权为$2$的Eisenstein级数构造，几何地推导出Dedekind-Rademacher同态及相应周期的经典公式。
In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state general conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative "period polynomials" in the case of cusp forms. We prove a special case of this conjecture in the case …
Prismatic crystals over the de Rham period sheaf
本文研究了离散赋值环棱晶景上的de Rham周期层晶体，利用对数联络和Kummer塔上的Sen-Fontaine理论，通过近de Rham表示对其进行分类，并比较了其棱晶上同调与Galois上同调。
This report outlines a combinatorial recipe for computing the bases, whose elements are oriented polygons, of two cohomology spaces associated to multizeta values: the top dimensional de Rham cohomology of moduli spaces of genus 0 complex pointed curves and the top dimensional de Rham cohomology of certain partial compactifications of these moduli spaces.
The main result of this paper is the existence of Galois representations associated with the mod $p$ (or mod $p^m$) cohomology of the locally symmetric spaces for $\GL_n$ over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for $\GL_n$ as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of …
This thesis is a study of algebraic and geometric relations between multizeta values. In chapter 2, we prove a result which gives the dimension of the associated depth-graded pieces of the double shuffle Lie algebra in depths 1 and 2. In chapters 3 and 4, we study geometric relations between multizeta values coming from their expression as periods on $\mathfrak{M}_{0,n}$. The key ingredient in this study is the top dimensional de Rham cohomology of special partially compactified moduli spaces associated to multizeta values. In chapter 3, we give an explicit expression for a basis, represented …
Periods of E-operators
该研究通过推广快速衰减上同调理论定义了E-周期，将含E-函数的绝对收敛积分实现为周期配对矩阵元，并将该上同调推广到奇异代数簇，证明了Nori基本引理的对应版本。
We consider the Tate cohomology of the circle group acting on the topological Hochschild homology of schemes. We show that in the case of a scheme smooth and proper over a finite field, this cohomology theory naturally gives rise to the cohomological interpretation of the Hasse-Weil zeta function by regularized determinants envisioned by Deninger. In this case, the periodicity of the zeta function is reflected by the periodicity of said cohomology theory, whereas neither is periodic in general.
Period-like polynomials for $L$-series associated with half-integral weight cusp forms
本研究构造了与半整权尖点形式$L$-级数对应的、系数在有限维向量空间的类Eichler上同调类，并定义了与两类形式$L$-级数相容的半整权到整权模形式的提升映射，推进了半整权模形式的上同调理论研究。
Cohomology classes, periods, and special values of Rankin-Selberg $L$-functions
本文对表示GL_n×GL_m与GL_n×GL_{n-m-1}的Rankin-Selberg L-函数乘积的积分给出上同调解释，并证明了特定L-函数特殊值的代数性，推广了Raghuram与Mahnkopf的相关结果。
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