# MA2317: Introduction to Number Theory

## Lecturer: Masha Vlasenko

We begin with the classical questions such as factorization, distribution of primes and finding integer solutions to algebraic equations. Second part of the course will deal with number fields and basics of algebraic number theory. Our ultimate goal will be to understand the quadratic reciprocity law and Dirichlet's theorem on arithmetic progressions in the context of class field theory. The last theme of the course is counting points on algebraic varieties over finite fields and the Weil conjectures. Though the statement of these conjectures is quite elementary, their proof took decades and had very high influence on the development of contemporary number theory.

The prerequisites are basic analysis, linear algebra and group theory. Lectures will be accompanied by numerous exercises intended to practice calculations with algebraic numbers, p-adic numbers and finite fields. There will be also bi-weekly tutorials in the form of problem-solving sessions.

In case one is motivated for further studies, we could recommend to look through the book of David A. Cox for higher reciprocity laws and to learn about the Birch and Swinnerton-Dyer conjecture. The latter concerns with the L-function of an elliptic curve which will be one of our last themes.

## Syllabus

1. Euclidean algorithm and Unique Factorization
• [IrelandRosen] chapter I
• Homework 1, due on October 7
2. Calculations with Residue Classes and the Quadratic Reciprocity Law
3. Some Modern Problems of Elementary Number Theory: Primality Tests, Factorization and Public Key Cryptosystems
• [Koblitz1] chapters III-V, [PanchishkinManin] chapter 2
• #### Vigenere Cipher Attack

This link will be activated at 12:00 on Friday, November 18th and will contain a ciphertext enciphered with the Vigenere algorithm: The file will contain a sequence of lowercase characters with spaces and punctuation marks. Decipher it and send me either the key or the plaintext (first line would suffice) to the address vlasenko@maths.tcd.ie. First successful submission gets + 15% to the final mark, next 4 get +10%.
Here is some stuff that might be helpful:
results

4. Distribution of Primes, Riemann Hypothesis, Primes in Arithmetic Progressions

5. Polynomials and Algebraic Extensions; Algebraic Numbers
6. Diophantine Equations of Degree Two and Class Numbers
• [Davenport] chapters V and VI, [Cox]
7. Irrationality and Diophantine Approximation
• [HardyWright] chapter IV, [PanchishkinManin] § 4 of chapter II
• [Koblitz2] chapter I
9. Cubic Diophantine Equations and Elliptic curves
• [Koblitz1] § 1 of chapter VI, [PanchishkinManin] § 3 of chapter 1 , [IrelandRosen] § 1 of chapter 18
10. Finite fields
• [Koblitz2] § 1 of chapter III, [IrelandRosen] chapter 7
11. Zeta function of an algebraic variety and Weil conjectures
• [Koblitz2] chapter V, [IrelandRosen] chapters 10, 11 and 18

## Literature

[IrelandRosen]
K.Ireland, M.Rosen, A Classical Introduction to Modern Number Theory
[Koblitz1]
N. Koblitz, A course in Number Theory and Cryptography
[Serre]
J.-P. Serre, A Course in Arithmetic
[PanchishkinManin]
Yu.I.Manin, A.A.Panchishkin, Introduction to Number Theory, Part I: Problems and Tricks
[Lang1]
S. Lang, Math Talks for Undergraduates
[Lang2]
S. Lang, Algebra
[Davenport]
H.Davenport, The Higher arithmetic
[Cox]
D.A. Cox, Primes of the form x^2 + n y^2
[HardyWright]
G.H.Hardy, E.M.Wright, An Introduction to the Theory of Numbers
[Koblitz2]
N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-Functions
[Chandrasekharan]
K. Chandrasekharan, Introduction to Analytic Number Theory

## Miscellaneous

PARI/GP calculator