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Introduction to Number Theory

010100.68

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PhD, , satoshi.kondo@gmail.com

_______________ 2015

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, 2015

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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 .

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 objectivesStudying 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: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 outcomesBy the end of the course student is supposed to

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:

(Algebra)

Course plan Total hoursContact hoursIndependent students workLecturesSeminars1

2





3

4

Quadratic reciprocity laws and the Hilbert symbols

The Riemann zeta function and the Dirichlet L-functions

Cyclotomic and quadratic extensions

The Dirichlet unit theoremVertex40

24

32

32 10

6

8

8

10

6

8

8

20

12

16

16

Total: 128 32 32 64

Requirements and GradingType of gradingType of work1 yearCharacteristics1 2 3 4 Running(week) Problem session2 2 Use blackboard8 midtermExaminationV wtitten, 120min.

FinalExaminationV wtitten, 120minKnowledge and skills grading criteriaAll work is graded on the scale from 1 to 10.

Calculating the grades for the course

The resulting grade for the running check evaluates the results of students work and is calculated according to the following formula:

= 0.9/ + 0.1 ;

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 is the evaluation of the students performance at the exam itself):

= 0,8 + 0,2This grade which is the resulting grade for the course is written down in the students certificate (diploma).

Content of the subjectSection 1. Quadratic reciprocity laws and the Hilbert symbols

TopicTotal hoursLecturesSeminarsIndependent students work1 Finite fields, the Legendre symbol 8 2 2 4

2 Conics over finite fields 8 2 2 4

3 p-adic numbers, the topology and the multiplicative structure 8 2 2 4

4 Conics over local fields 8 2 2 4

5 The Hasse principle 8 2 2 4

Total: 40 10 10 20

Section 2. the Riemann zeta function and the Dirichlet L-function

TopicTotal hoursContact hoursIndependent students workLecturesSeminars1 General L-series 8 2 2 4

2 The analytic continuation using Hurwitz zeta 8 2 2 4

3 The special values of the Riemann zeta function 8 2 2 4

Total: 24 6 6 12

Section 3. Abelian extensions TopicTotal hoursContact hoursIndependent students workLecturesSeminars1 Cyclotomic fields and quadratic fields 8 2 2 4

2 Decomposition of primes in an extension 8 2 2 4

3 The Galois group of a cyclotomic field 8 2 2 4

4 Class field theory and quadratic reciprocity 8 2 2 4

Total: 32 8 8 16

Section 4. Algebraic number theory TopicTotal hoursContact hoursIndependent students workLecturesSeminars1 Local and global fields8 2 2 4

2 Adeles and ideles8 2 2 4

3 The Dirichlet unit theorem8 2 2 4

4 The finiteness of class number 8 2 2 4

Total: 32 8 8 16

Grading estimation for the running check and the final assessment of students

Topics for Current ControlQuestion for the test:

Compute the Legendre symbols and the Hilbert symbols.

Compute the p-adic expansions of exponential or logarithm functions.

Verify the topological properties of the p-adic integers.

Construct abelian field extensions with prescribed ramifications.

Compute the ramification in quadratic extensions.

Apply the Dirichlet unit theorem to compute the group of units.

Questions for evaluating students performance

Describe the metric in nonarchimedean local fields.

Determine if a conic has a solution in the field of rational numbers or in local fields.

Describe the analytic properties of the Riemann zeta function and the Dichlet L-functions.

Find the primitive roots modulo a prime number.

Determine the Galois group of some subfields of cyclotomic fields.

State and give an application of the finiteness theorem of idele class groups.

Relate the group of units in the integer ring and K-theory.

Give an explanation to the special values of zeta functions.

Readings and materials for the course

Fundamental textbookKato, 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

Required readingManin, 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

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

Algebraic Number Theory ed. Cassels and Frohlich- London Mathematical Society; 2 edition (March 12, 2010) ISBN-10: 0950273422ISBN-13: 978-0950273426


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