Intermediate extensions and crystalline distribution algebras

Intermediate extension of Chow motives of Abelian type

In this article, we give an unconditional construction of a motivic analogue of the intermediate extension in the context of Chow motives of Abelian type. Our main application concerns intermediate extensions of Chow motives associated to Kuga families to the Baily--Borel compactification of a Shimura variety.

1 的中间扩张、消没圈与反常特征空间

The Intermediate Extension, Vanishing Cycles, and Perverse Eigenspace of One

本文证明了关于消没圈上恒等映射减去单径核的若干结果。

Syntactic presentations for glued toposes and for crystalline toposes

We regard a geometric theory classified by a topos as a syntactic presentation for the topos and develop tools for finding such presentations. Extensions of geometric theories, which can add axioms, symbols and sorts, are treated as objects in their own right, to be able to build up complex theories from parts. The role of equivalence extensions, which leave the theory the same up to Morita equivalence, is investigated. Motivated by the question what the big Zariski topos of a non-affine scheme classifies, we show how to construct a syntactic presentation for a topos if syntactic presentations…

A motivic conjecture of Milne

Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, the étale cohomology with $\dbZ_p$ coefficients to the crystalline cohomology with integral coefficients, in the more general context of p-divisible groups endowed with {\it arbitrary} families of crystalline tensors over a finite, discrete valuation ring extension of W(k). This extends a result of Faltings in [Fa2]. As a main new tool we construct global deformations of p-divisible groups…

仿射Grassmannian中的圈关系及其在G-晶体表示的Breuil--Mézard中的应用

Cycles relations in the affine grassmannian and applications to Breuil--Mézard for G-crystalline representations

本文通过仿射Grassmannian中的闭子概形圈关系实现了$\widehat{G}$-表示的Grothendieck群恒等式，并将其与$G$-值晶体表示模空间几何相联系，推导出小Hodge类型下Breuil--Mézard猜想的一个方向。

Sur les extensions intermédiaires des systèmes locaux d'Harris-Taylor

In the geometric situation of some simple unitary Shimura varieties studied by Harris and Taylor, I have built two filtrations of the perverse sheaf of vanishing cycles. The graduate of the first are the $p$-intermediate extension of some local Harris-Taylor's local systems, while for the second, obtained by duality, they are the $p+$-intermediate extensions. In this work, we describe the difference between these $p$ and $p+$ intermediate extension. Precisely, we show, in the case where the local system is associated to an irreducible cuspidal representation whose reduction modulo $l$ is super…

Intermediate extensions of perverse constructible $\mathbb{F}_p$-sheaves commute with smooth pullbacks

We prove that intermediate extensions of perverse constructible $\mathbb{F}_p$-sheaves commute with smooth pullbacks for schemes admitting a closed embedding into a smooth scheme over a field of characteristic $p$ (embeddable schemes for short). Along the way we also prove that the equivalence of categories of Cartier crystals with unit $R[F]$-modules commutes with $f^!$ for a smooth morphism $f: X \to Y$ of embeddable schemes.

Arithmetic of Chatelet surfaces under extensions of base fields

For Châtelet surfaces defined over number fields, we study two arithmetic properties, the Hasse principle and weak approximation, when passing to an extension of the base field. Generalizing a construction of Y. Liang, we show that for an arbitrary extension of number fields $L/K,$ there is a Châtelet surface over $K$ which does not satisfy weak approximation over any intermediate field of $L/K,$ and a Châtelet surface over $K$ which satisfies the Hasse principle over an intermediate field $L'$ if and only if $[L' : K]$ is even.

除幂与导出德拉姆上同调

Divided Powers and Derived De Rham Cohomology

本研究构建导出除幂代数形式体系，在该框架下重新研究导出德拉姆与晶体上同调，给出其泛性质刻画，验证其与经典理论的一致性，并推导相关代数几何工具的应用。