Tits Geometry, Arithmetic Groups and the Proof of a Conjecture of Siegel

Geometry and Arithmetic on the Siegel-Jacobi Space

The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space.

Green forms and the arithmetic Siegel-Weil formula

We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p,2) and U(p,1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.

Siegel-Jacobi 空间的几何问题

Problems in the Geometry of the Siegel-Jacobi Space

本文针对非对称齐次空间 Siegel-Jacobi 空间，提出其几何中的基本问题，强调其在几何与算术中的重要性。

A note on invariant differential operators on Siegel-Jacobi space

In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential operators. We present problems which are natural. We give some new partial solutions for these natural problems.

Robert MacPherson and arithmetic groups

We survey contributions of Robert MacPherson to the theory of arithmetic groups. There are two main areas we discuss: (i) explicit reduction theory for Siegel modular threefolds, and (ii) constructions of compactifications of locally symmetric spaces. The former is joint work with Mark McConnell, the latter with Lizhen Ji.

算术双曲流形中的纽结数与特殊微分特征的关联

Linking numbers and special differential characters in arithmetic hyperbolic manifolds

该研究构造了编码最简型算术双曲流形纽结数的非全纯Siegel模形式，通过深化对Kudla–Millson理论中上迁形式的理解完成证明，并建立了相关结构性质。