This course deals with reciprocity for algebraic curves over number fields, and provides the basics of algebraic K-groups for schemes. The instructor will first review homotopy theory for topology, then explain the definition and properties of algebraic K-groups, investigating several examples of low-order algebraic K-groups. We will then study the relationship between algebraic K-groups on fields and étale cohomology. Using this, students will construct a reciprocity map for algebraic curves on a number field, and prove reciprocity. This course follows Advanced Topics in Algebra G.
Algebraic K-groups are useful for answering several questions in the geometry of numbers. This course is used for investigating algebraic K-groups and étale cohomology with a reciprocity map of algebraic curves on a number field.
By the end of this course, students will be able to:
1) Understand the definition and some of the basic properties of the algebraic K-groups of schemes.
2) Understand the structure of the 0th algebraic K-groups and the low-dimensional algebraic K-groups of fields.
3) Understand the relation between the algebraic K-groups and the étale cohomologies of fields.
4) Understand the reciprocity law of an algebraic curve over a number field.
homotopy group, algebraic K-group, etale cohomology, algebraic cycle, reciprocity law
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 to the classical class field theory | Details will be provided during each class session. |
Class 2 | the abstract of the algebraic K-theory, and the homotopy groups of categories | Details will be provided during each class session. |
Class 3 | the definition and some of the basic properties of algebraic K-groups of schemes | Details will be provided during each class session. |
Class 4 | the low-dimensional algebraic K-groups, and the algebraic K-groups of fields | Details will be provided during each class session. |
Class 5 | the relation between the algebraic K-group and the étale colomology of fields | Details will be provided during each class session. |
Class 6 | the construction of the reciprocity map | Details will be provided during each class session. |
Class 7 | the reciprocity law for the algebraic curves over local fields | Details will be provided during each class session. |
Class 8 | the reciprocity law for the algebraic curves over number fields | Details will be provided during each class session. |
Unspecified.
Course materials are provided during class.
Learning achievement is evaluated by reports.
Students must have successfully completed Algebra I, Algebra II , Algebra III and Advanced topics in Algebra G;
or, they must have equivalent knowledge.