一、 課程背景與目的：

Due to recent very active development of arithmetic of Shimura varieties, more people are interested in computing the numbers of connected and irreducible components of certain subvarieties of Shimura varieties make impressive progress in this direction. Mass formula is one of useful tools for computing the derived arithmetic invariants. In this course we will revisit the works of Shimura, Gross, Gan  JK Yu and others, and discuss related working knowledge in the field. 

    

二、課程之大綱與講者：

We will introduce the Tamagawa measure and mass for a reductive group. 

Our focus will be mainly on the quaternion unitary groups and unitary groups. 

We shall first describe the main results of [5]. The main part of this course is to report the results of the papers [3], [1] and [2]. 

If time permits we will describe Shimura’s mass formula in [4] and compare the improvement made in [1] and [2]. 

 

References:

[1] W. T. Gan, J. P. Hanke, and J.-K. Yu, On an exact mass formula of Shimura. 

Duke Math. J. 107 (2001), 103--133.

[2] W. T. Gan and J.-K. Yu, Group schemes and local densities. 

Duke Math. J. 105 (2000), 497--524.

[3] B. H. Gross, On the motive of a reductive group. 

Invent. Math. 130 (1997), 287--313.

[4] G. Shimura, Euler product and Eisenstein series. 

CBMS Regional Conference Series in Math., 93, Washington, DC, 1997.

[5] G. Shimura, Some exact formulas on quaternion unitary groups. 

J. Reine Angew. Math. 509 (1999), 67–102.

 

No class on 12/16, 12/23. Make-up classes on 1/15 & 1/22.

 

Time change on 1/27 Course: from 1:50-3:20 to 4:10-5:40.