This course deals with the unramified class field theory of algebraic surfaces over finite fields, and provides the basics of fundamentals groups and Chow groups of schemes. The instructor will review Picard groups of schemes, then explain the definitions and properties of algebraic K-groups and Chow groups, investigating several examples. Next, the instructor define the fundamental groups of schemes and express them using étale cohomologies. We will then study the relationship between algebraic K-groups on fields and étale cohomology. Using this, students will construct reciprocity maps for algebraic varieties over finite fields, and prove the main theorem of the unramified class field theory. This course follows Advanced Topics in Algebra E.
Algebraic K-groups are useful for answering several questions in the arithmetic geometry. This course is used for investigating algebraic K-groups and étale cohomologies with reciprocity maps of algebraic varieties over finite fields.
By the end of this course, students will be able to:
1) Understand the analogy between integer rings and algebraic curves over finite fields.
2) Understand étale cohomologies of algebraic curves over algebraically closed fields.
3) Understand the structure of algebraic K-groups of fields.
4) Understand the unramified class field theory of algebraic curves over finite fields.
étale cohomologies, fundamental groups, Chow groups, algebraic K-groups, unramified class field theory
Intercultural skills | Communication skills | ✔ Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course.
Course schedule | Required learning | |
---|---|---|
Class 1 | introduction | Details will be provided during each class session. |
Class 2 | étale cohomologies of algebraic curves over algebraically closed fields | Details will be provided during each class session. |
Class 3 | the fundamental groups of schemes | Details will be provided during each class session. |
Class 4 | the Picard groups and the Chow groups of schemes | Details will be provided during each class session. |
Class 5 | unramified class field theory of algebraic curves over finite fields | Details will be provided during each class session. |
Class 6 | algebraic K-groups of fields and the conjecture of Bloch and Kato | Details will be provided during each class session. |
Class 7 | the construction of the reciprocity maps | Details will be provided during each class session. |
Class 8 | unramified class field theory of algebraic surfaces over finite fields | Details will be provided during each class session. |
Unspecified.
Course materials are provided during class.
Learning achievement is evaluated by reports(100%).
Students must have successfully completed Algebra I, Algebra II , Algebra III and Advanced topics in Algebra E.