Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Introduction to Number Theory» для направления
01.04.01 «Математика» подготовки магистра
Правительство Российской Федерации
Федеральное государственное автономное образовательное учреждение высшего
профессионального образования
"Национальный исследовательский университет
"Высшая школа экономики"
Факультет Математики
Программа дисциплины
Introduction to Number Theory
для направления 010100.68 «Математика» подготовки магистра
магистерская программа «Математика» (англоязычная)
Автор программы:
PhD, Кондо Сатоши, satoshi.kondo@gmail.com
Рекомендована секцией УМС по математике «___»____________ 2015 г
Председатель С.М. Хорошкин
Утверждена УС факультета математики «___»_____________2015 г.
Ученый секретарь Ю.М. Бурман_______________________________
Москва, 2015
Настоящая программа не может быть использована другими подразделениями университета
и другими вузами без разрешения кафедры-разработчика программы.
Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Introduction to Number Theory» для направления
01.04.01 «Математика» подготовки магистра
1
Scope of use and legal references
This course program sets minimal requirements to the knowledge and skills of students and determines
the contents and kinds of lectures and reporting. The program is intended for lecturers teaching this course,
course assistants and students of 01.04.01 specialization «Mathematics» who study the course
“Introduction to Number Theory ”.


2
The program has been elaborated in accordance with the Educational standard of HSE for training
area 01.04.01 «Mathematics» (Master level);
In accordance with the working studying plan of the university for training area of 01.04.01 specialization «Mathematics» (Master level), master program “Mathematics”, approved in 2014.
Learning objectives
Studying the course “Introduction to Number Theory ” aims at making the students familiar with the main
examples, constructions and techniques of the theory of infinite dimensional Lie algebras with applications
in mathematical physics. In particular, the students will learn the basics of the theory of vertex operator algebras, playing the crucial role in the modern mathematical language of the quantum filed theory.
Course goals:






3
To make students familiar with the simpler examples of the Diophantine questions.
To describe the Hasse principle via the example of quadratic form in two variables.
To study the Quadratic reciprocity and perform computation of the Legendre symbols.
To make clear the role of the Riemann zeta function and the Dirichlet L-functions in arithmetic geometry.
To educate students about major concepts in algebraic number theory.
To expose students to various broad subjects in number theory.
Learning outcomes
By the end of the course student is supposed to


4
Know: The basic concepts in algebra and algebraic number theory such as zeta functions, local
and global fields.
Be able to: Understand some basic terminology in modern number theory. Read without help the
more advanced texts aimed at graduate students. Solve or perform example computations when
given problems in number theory.
Place of Discipline in MA program structure
This course is a professional one. This is an elective course for the “Mathematics” specialization.
This course is based on knowledge and competences that were provided by the following disciplines:

Алгебра (Algebra)
Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Introduction to Number Theory» для направления
01.04.01 «Математика» подготовки магистра
5
Course plan
№
1
2
3
4
6
Total
hours
Название раздела
Quadratic reciprocity laws and the Hilbert
symbols
The Riemann zeta function and the Dirichlet Lfunctions
Cyclotomic and quadratic extensions
The Dirichlet unit theoremVertex
40
Total:
Contact hours
Lecture
Seminars
s
10
10
Independent
student’s work
20
24
6
6
12
32
32
8
8
8
8
16
16
128
32
32
64
Requirements and Grading
Type of grading
Type of work
Running
(week)
Problem
session
1 year
1 2 3
4
2 2
Characteristics
Use blackboard
8
midterm
Final
6.1
Examination
Examination
V
V
wtitten, 120 min.
wtitten, 120 min
Knowledge and skills grading criteria
All work is graded on the scale from 1 to 10.
6.2
Calculating the grades for the course
The resulting grade for the running check evaluates the results of student’s work and is calculated according to the following formula:
Отекущий = 0.9·Ок/р + 0.1·Одз ;
A student is required to participate in the problem sessions. He/she needs to meet the requirement to be
eligible to take the examinations. The resulting grade for the final evaluation in the form of examination is
calculated according to this formula (where Оэкзамен is the evaluation of the student’s performance at the
exam itself):
Оитоговый = 0,8·Оэкзамен + 0,2·Отекущий
This grade which is the resulting grade for the course is written down in the student’s certificate (diploma).
Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Introduction to Number Theory» для направления
01.04.01 «Математика» подготовки магистра
7
Content of the subject
Section 1. Quadratic reciprocity laws and the Hilbert symbols
№
Topic
1
Finite fields, the Legendre symbol
2
Conics over finite fields
3
4
p-adic numbers, the topology and the multiplicative structure
Conics over local fields
5
The Hasse principle
Total:
Total
hour
s
Lectures
Seminars
Independent
student’s work
8
2
2
4
8
2
2
4
8
2
2
4
8
2
2
4
8
2
2
4
40
10
10
20
Section 2. the Riemann zeta function and the Dirichlet L-function
№
1
2
3
Topic
General L-series
The analytic continuation using Hurwitz zeta
The special values of the Riemann zeta function
Total:
Total
hours
8
8
8
24
Contact hours
Lecture
Seminars
s
2
2
2
2
2
2
6
6
Independent
student’s work
4
4
4
12
Section 3. Abelian extensions
№
1
2
3
4
Topic
Cyclotomic fields and quadratic fields
Decomposition of primes in an extension
The Galois group of a cyclotomic field
Class field theory and quadratic reciprocity
Total:
Section 4. Algebraic number theory
Total
hours
8
8
8
8
32
Contact hours
Lecture
Seminars
s
2
2
2
2
2
2
2
2
8
8
Independent
student’s work
4
4
4
4
16
Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Introduction to Number Theory» для направления
01.04.01 «Математика» подготовки магистра
№
1
2
3
4
8
Topic
Local and global fields
Adeles and ideles
The Dirichlet unit theorem
The finiteness of class number
Total:
Total
hours
8
8
8
8
32
Contact hours
Lecture
Seminars
s
2
2
2
2
2
2
2
2
8
8
Independent
student’s work
4
4
4
4
16
Grading estimation for the running check and the final assessment of students
8.1
Topics for Current Control
Question for the test:
1. Compute the Legendre symbols and the Hilbert symbols.
2. Compute the p-adic expansions of exponential or logarithm functions.
3. Verify the topological properties of the p-adic integers.
4. Construct abelian field extensions with prescribed ramifications.
5. Compute the ramification in quadratic extensions.
6. Apply the Dirichlet unit theorem to compute the group of units.
8.2
Questions for evaluating student’s performance
1. Describe the metric in nonarchimedean local fields.
2. Determine if a conic has a solution in the field of rational numbers or in local fields.
3. Describe the analytic properties of the Riemann zeta function and the Dichlet L-functions.
4. Find the primitive roots modulo a prime number.
5. Determine the Galois group of some subfields of cyclotomic fields.
6. State and give an application of the finiteness theorem of idele class groups.
7. Relate the group of units in the integer ring and K-theory.
8. Give an explanation to the special values of zeta functions.
9
Readings and materials for the course
Fundamental textbook
Kato, Kazuya; Kurokawa, Nobushige; Saito, Takeshi Number theory. 2. Introduction to class field
theory. Translated from the 1998 Japanese original by Masato Kuwata and Katsumi Nomizu. Translations
of Mathematical Monographs, 240. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2011. viii+240 pp. ISBN: 978-0-8218-1355-3
9.1
Required reading
Manin, Yuri Ivanovic; Panchishkin, Alexei A. Introduction to modern number theory. Fundamental
problems, ideas and theories. Translated from the Russian. Second edition. Encyclopaedia of Mathematical
Sciences, 49.Springer-Verlag, Berlin, 2005. xvi+514 pp. ISBN: 978-3-540-20364-3; 3-540-20364-8
9.2
Национальный исследовательский университет «Высшая школа экономики»
Программа дисциплины «Introduction to Number Theory» для направления
01.04.01 «Математика» подготовки магистра
Serre, J.-P. A course in arithmetic. Translated from the French. Graduate Texts in Mathematics, No.
7.Springer-Verlag, New York-Heidelberg, 1973. viii+115 pp
Further reading
Kato, Kazuya; Kurokawa, Nobushige; Saito, Takeshi Number theory. 1. Fermat's dream. Translated
from the 1996 Japanese original by Masato Kuwata. Translations of Mathematical Monographs, 186.
Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2000. xvi+154
pp. ISBN: 0-8218-0863-X
9.3
Algebraic Number Theory ed. Cassels and Frohlich- London Mathematical Society; 2 edition
(March 12, 2010) ISBN-10: 0950273422ISBN-13: 978-0950273426