Nonhomeomorphic conjugates of connected Shimura varieties

Galois conjugates of special points and special subvarieties in Shimura varieties

Let $S$ be a Shimura variety with reflex field $E$. We prove that the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih's construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne.

An example of non-homeomorphic conjugate varieties

We give examples of smooth quasi-projective varieties over complex numbers, in the context of connected Shimura varieties, which are not homeomorphic to a conjugate of itself by an automorphism of the complex numbers.

Admissible morphisms for Shimura varieties with parahoric levels

In \textit{Shimuravarietäten und Gerben} \cite{LR87}, Langlands and Rapoport developed the theory of pseudo-motivic Galois gerb and admissible morphisms between Galois gerbs, with a view to formulating a conjectural description of the $\bar{\mathbb{F}}_p$-point set of the good reduction of a Shimura variety with hyperspecial level, as well as to providing potential tools for its resolution. Here, we generalize, and also improve to some extent, their works to parahoric levels when the group is quasi-split at $p$. In particular, we show that every admissible morphism is conjugate to a special ad…

On the cohomology of compact unitary group Shimura varieties at ramified split places

In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke operators at p on the automorphic side. We allow arbitrary ramification at p; even the PEL data may be ramified. This gives a description of the semisimple local Hasse-Weil zeta function in these cases. We also treat cases of nontrivial endoscopy. For this purpose, we give a general stabilization of the expression given in previous work, following the stabilization…

Zip 周期映射的 Ogus 原理：通过“Frobenius 与 Hodge 滤过”研究 Hasse 不变量的消失阶

An Ogus Principle for Zip period maps: The Hasse invariant's vanishing order via `Frobenius and the Hodge filtration'

本文将 Ogus 关于 Calabi–Yau 族 Hasse 不变量消失阶与共轭线位置的结果推广到群论 Zip 周期映射，并导出 Shimura 簇及 K3 模空间上的原理。

Virtual Abelian varieties of $\mathrm{GL}_2$-type

This paper studies a class of Abelian varieties that are of $\GL_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (1) a totally real field $K$ or (2) a totally indefinite quaternion algebra over a totally real field $K$. Among the isogenous class of such an Abelian variety, we identify one whose Galois conjugates can be described in terms of actions of Atkin-Lehner operators and the class group of $K$. Thus we deduce that such Abelian varieties are parametrised by finite quotients of certain PEL Shimura varieties…