Research School

 

WAMS Research School

Topics in Analytic and Transcendental
Number Theory

Institute for Advanced Studies in Basic Sciences (IASBS)
Zanjan, Iran - July 1th - July 13th 2017


General Information

Topics :

  1. Some basic non complex methods in analytical number theory
  2. Diophantine Approximation
  3. Analytic Problems for Elliptic Curves

 

  • Courses open to students with different backgrounds and levels
  • Each of the three courses will include tutorials
  • Each course will consist of ten lectures of 90 minutes each, four sessions out of the ten lectures being devoted to exercises and problem solving
  • Some funds are available to cover (in part) the travel and local expenses for International and Iranian participants

Application Procedure

1.International applicants should fill the following registration form
APPLY TO THE WAMS SCHOOL

2.Iranian applicants should fill the following form
APPLY TO THE WAMS SCHOOL

Deadline for applications: April 15th, 2017
Final decision for acceptance and funds offers: April 25th, 2017
All applicants should arrange for a presentation letter submitted, before the deadline, to zanjan2017@rnta.eu
The criteria for selection of the applicants will follow those of CIMPA research school.

Courses

Elementary methods in analytic number theory
Lecturers: Mehdi Hassani & Jean-Marc Deshouillers

Part I. Elementary and combinatorial number theory

  • Generalities on primes
  • Arithmetical functions
  • Dirichlet convolution
  • Formal Euler products
  • Mobius inversion formulas
  • Chebyshev's upper and lower bounds for primes up to X.
  • Brun's sieve and twin primes

Part II. Calculus

  • Mean value of some arithmetical functions
  • Divisors and the hyperbola
  • Sums of two squares and the circle problem

Part III. Fourier

  • Upper bounds for trigonometric sums : van der Corput
  • Application to Voronoi's theorem
  • Application to the circle problem

Part IV. Heuristics and probabilistic methods (Time permitting)

  • Normal orders of arithmetic functions
  • Distribution modulo 1

REFERENCES:
Gerald Tenenbaum Course on analytic and probabilistic number theory (published in English by AMS) is as a very good reference for the above course and related topics.
 

A course on Diophantine Approximation
Lecturer: Jean-Marc Deshouillers & Sanoli Gun

Part I : Introduction to Diophantine Approximation:

  • Dirichlet's theorem
  • Application to Pell's equation
  • Liouville's theorem and irrationality measure
  • Liouville numbers and a theorem of Erdős
  • Khintchine's theorem
  • Irrationality of some classical numbers, say e, e2, π, ζ(3)

Part II : Continued Fractions:

  • Basics of theory of continued fractions
  • Farey sequences and their equidistribution
  • Badly approximable numbers and Littlewood conjecture
  • Application to quadratic irrationals

Part III : Deeper Diophantine theorems:

  • Thue-Siegel-Roth-Ridout theorems (statements)
  • Application to Diophantine equations;
  • Warings's problem, Thue's equations and elementary applications to frequency of digits in base-b expansions.
  • p-adic subspace theorem (statement) and deeper application to complexity of algebraic irrationals.

Analytic Problems for Elliptic curves

Lecturers: Amir Akbary & Francesco Pappalardi
Part I : Algebraic Number Theory

  • basics of algebraic number theory (Number fields, rings of integers, splitting of primes)
  • splitting of primes in cyclotomic fields and primes in arithmetic progressions

Part II : Elliptic Curves

  • basics of elliptic curves (structure of the group of rational points, elliptic curves over finite fields)
  • reduction mod p of elliptic curves
  • splitting of primes in division fields of elliptic curves

Part III : Analytic Results

  • Prime number theorem for arithmetic progressions
  • Bombieri-Vinogradov theorem
  • Chebotarev density theorem

Part IV : Applications

  • classical Titchmarch divisor problem
  • elliptic Titchmarsh divisor problem
  • Serre's cyclicity problem
  • Artin's conjecture and its elliptic analogue (time permitting)

REFERENCES:
Lawrence C. Washington, Elliptic Curves: Number Theory and Crptography. Chapman & Hall (CRC) 2003. Joseph
H. Silverman, The Arithmetic of Elliptic Curves. Springer GTM 2009
Alina Carmen Cojocaru, Questions about the reductions modulo primes of an elliptic curve. Number theory, 61-79, CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI, 2004


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