一、課程背景與目的：

Algebraic tori and unitary groups appears in many areas of mathematics, from the various aspect: analytic, algebraic and arithmetic. They have been studied intensively and there are still many unanswered questions. The goal of this course is to introduce arithmetic theory of algebraic tori and Hermitian forms and introduce the students recent results on the class numbers and Tamagawa numbers of certain tori and mass and class numbers of certain unitary groups. 

 

二、課程之大綱：

We divide the course into 4 parts

1: Prerequisite from algebraic number theory. Local and global fields, trace, norm, different, discriminant, regulators, class number formulas, local symbols, local and global indices, CM and RM fields, class number relations

2: Algebraic groups and their arithmetic: algebraic tori, reductive groups, class numbers and Tamagawa numbers, Galois cohomology,

3: Hermitian spaces, Hermitian lattices, class numbers of non-definite Hermitian lattices, mass formulas, connected components of unitary Shimura varieties.