TOKYO INSTITUTE OF TECHNOLOGY

Course description and aims

Anabelian geometry, which is an area of arithmetic geometry, was proposed by A. Grothendieck in the 1980's based on his intuition that the geometric information of an anabelian variety is completely determined by the purely group-theoretic property of the algebraic fundamental group of the anabelian variety. One main purpose of this lecture is to provide an introduction to anabelian geometry by focusing on anabelian geometry of algebraic number fields and mixed-characteristic local fields, which play central roles in the study of number theory. This lecture will explain various group-theoretic reconstruction algorithms in mono-anabelian geometry whose input data consist of topological groups isomorphic to the absolute Galois groups of such fields and whose output data consist of some arithmetic invariants of such fields.
One may find that several fundamental ideas and methods of anabelian geometry appear in discussions of the algorithmic approach to anabelian geometry of mixed-characteristic local fields. Thus, it seems to me that the algorithmic approach to anabelian geometry of mixed-characteristic local fields may be regarded as one suitable topic for introduction to anabelian geometry. Moreover, one may also find that various important arithmetic theories, such as Kummer theory, local class field theory, and global class field theory, will be applied in an essential way in this lecture. It seems to me that it is meaningful, from an educational point of view, to have an opportunity of applying such important arithmetic theories.

Student learning outcomes

To understand the following issues:
fundamental facts concerning mixed-characteristic local fields and their absolute Galois groups
constructions of mono-anabelian reconstruction algorithms related to mixed-characteristic local fields
fundamental facts concerning algebraic number fields and their absolute Galois groups
constructions of mono-anabelian reconstruction algorithms related to algebraic number fields

Keywords

anabelian geometry, algebraic number field, mixed-characteristic local field, absolute Galois group, mono-anabelian reconstruction algorithm, local-global cyclotomic synchronization

Competencies that will be developed

Class flow

This is a standard lecture course.

Course schedule/Required learning

Textbook(s)

None required

Reference books, course materials, etc.

Will be announced in the class.

Assessment criteria and methods

Assignments (100%).

Related courses

• MTH.A301 ： Algebra I
• MTH.A302 ： Algebra II
• MTH.A331 ： Algebra III

Prerequisites (i.e., required knowledge, skills, courses, etc.)

None required