Counting integral points on some homogeneous varieties with large reductive stabilizers

A refinement of the Hodge stratification for connected reductive groups

For connected reductive groups G over a finite extension F of Q_p and L the maximal unramified extension of F we study the sets H_{μ, N}(G) of elements b in G(L) with given Hodge points of (bσ), (bσ)^2, ..., (bσ)^N. We explain the relationship to stratifications of some moduli scheme of abelian varieties defined by Goren and Oort respectively Andreatta and Goren. We show that for sufficiently large N the Newton point is constant on the sets H_{μ, N}(G) and compute such N for certain classes of groups.

Shimura 曲线的非约化循环与扭曲算术转移

Non-reductive cycles and twisted arithmetic transfers for Shimura curves

本文综述 Shimura 簇和 Rapoport-Zink 空间新循环、扭曲算术基本引理及相对 Langlands 算术类比的进展，阐明算术扭曲 GGP 猜想表述、扭曲 AFL 证明与相关算术转移。

半阿贝尔簇的几何Bogomolov猜想

Geometric Bogomolov conjecture for semiabelian varieties

本文证明了函数域上半阿贝尔簇的几何Bogomolov猜想，刻画了闭子簇包含Zariski稠密小点集的充要条件，并揭示了特殊子簇可能缺乏高度为0点的Zariski稠密集这一新现象。

有理性型affine Deligne-Lusztig变种的连通分量

The connected components of affine Deligne--Lusztig varieties

本文通过p-adic Hodge理论方法，全面解决了affine Deligne-Lusztig变种和局部Shimura变种的连通分量问题，并推导了关于Shimura变种积分模型的新几何结果。

Regular integral models for Shimura varieties of orthogonal type

We consider Shimura varieties for orthogonal or spin groups acting on hermitian symmetric domains of type IV. We give regular p-adic integral models for these varieties over odd primes p at which the level subgroup is the connected stabilizer of a vertex lattice in the orthogonal space. Our construction is obtained by combining results of Kisin and the first author with an explicit presentation and resolution of a corresponding local model.

p进函数域上强逼近的互反障碍

Reciprocity obstruction to strong approximation over p-adic function fields

本文在p进曲线函数域上构造SL_n齐性空间反例，证明强逼近失效，并提出且刻画了作为唯一障碍的互反障碍。

可分离伪可约带及其在有理点应用

Separable Pseudo-reductive Bands with Applications to Rational Points

本文扩展了Galois理论中关于代数带的Borovoi-Springer解释，研究可分离带在任意字段的fppf位势上局部表示的性质，并证明在特定条件下的全局表示性，同时推广了Borovoi的阿贝尔化理论并应用到有理点问题。

The $p$-adic monodromy group of abelian varieties over global function fields of characteristic $p$

We prove an analogue of the Tate isogeny conjecture and the semi-simplicity conjecture for overconvergent crystalline Dieudonné modules of abelian varieties defined over global function fields of characteristic $p$. As a corollary we deduce that monodromy groups of such overconvergent crystalline Dieudonné modules are reductive, and after a finite base change of coefficients their connected components are the same as the connected components of monodromy groups of Galois representations on the corresponding $l$-adic Tate modules, for $l$ different from $p$. We also show such a result for gener…

Cordial elements and dimensions of affine Deligne-Lusztig varieties

The affine Deligne-Lusztig variety $X_w(b)$ in the affine flag variety of a reductive group $\mathbb G$ depends on two parameters: the $σ$-conjugacy class $[b]$ and the element $w$ in the Iwahori-Weyl group $\tilde W$ of $\mathbb G$. In this paper, for any given $σ$-conjugacy class $[b]$, we determine the nonemptiness pattern and the dimension formula of $X_w(b)$ for most $w \in \tilde W$.

On the dimension of some union of affine Deligne-Lusztig varieties

In this paper, we consider certain union $X(μ,b)$ of affine Deligne-Lusztig varieties in the affine flag variety that arises in the study of mod-$p$ reduction of Rapoport-Zink spaces and moduli spaces of shtukas associated to a connected reductive group. Under a mild hypothesis on $μ$, but no further restrictions on the group, we compute its dimension in the case where $b$ is the maximal neutrally acceptable element.

Compactly supported cohomology and nearby cycle cohomology of open Shimura varieties of PEL type

In this paper, we compare two cohomology groups associated to Shimura varieties of PEL type, which are not proper over the base. One is the compactly supported l-adic cohomology, and the other is the nearby cycle cohomology, namely, the compactly supported cohomology of the nearby cycle complex for the canonical integral model of the Shimura variety over Z_p. We prove that the G(Q_p)-cuspidal part of these cohomology groups are the same, where G denotes the reductive algebraic group naturally attached to the PEL datum. Some applications to unitary Shimura varieties are also given.

Affine Deligne-Lusztig Varieties and Quantum Bruhat Graph

In this paper, we consider affine Deligne-Lusztig varieties $X_w(b)$ and their certain union $X(μ,b)$ inside the affine flag variety of a reductive group. Several important results in the study of affine Deligne-Lusztig varieties have been established under the so-called superregularity hypothesis. Such results include a description of generic Newton points in Iwahori double cosets of loop groups, covering relation in associated Iwahori-Weyl group and dimension formula for $X(μ,b)$. We show that one can considerably weaken the superregularity hypothesis and sometimes completely eliminate it, t…

分歧素数上某些酉 Shimura 簇的半稳定模型

Semi-stable models for some unitary Shimura varieties over ramified primes

本文针对签名 $(2,n-2)$ 的酉群相关的 Shimura 簇，通过显式分解局部模型，在虚二次域分歧的奇素数 $p$ 上构建了正则 $p$-adic 整模型。

Affine Deligne-Lusztig varieties at infinite level

We initiate the study of affine Deligne-Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for GLn and its inner forms, Lusztig's semi-infinite Deligne-Lusztig construction is isomorphic to an affine Deligne-Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet--Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig's 1979 conjecture in this set…