A CIMPA research school on
Explicit Number Theory
The University of the Witwatersrand
Johannesburg, South Africa
January 8th-19th 2018

Courses


Introduction to analytic number theory
Florian Luca


  1. Chebyshev estimates;
  2. Abel summation formula;
  3. The Moebius function;
  4. The prime number theorem;
  5. Mean values of arithmetic functions;
  6. The Brun pure sieve.
Lecture 1, Lecture 5,
Lecture 2, Lecture 6,
Lecture 3, Lecture 7
Lecture 4,
Introduction to algebraic number theory
Valerio Talamanca


  1. Number fields
  2. Norms, traces, and discriminants
  3. Ring of integers
  4. Ideal factorization in Dedekind rings
  5. Decomposition and ramification
  6. Archimedean and non archimedean absolute values
Field and Galois theroy, by J.Milne
Algebraic numbr field, by J. Milne
Number Theory, by R. Schoof
Number rings, by P. Stevenhagen
Explicit solution of Diophantine equations
Yuri Bilu


  1. Classical Diophantine equations in two variables, their theory, relations with Diophantine Approximation and with Algebraic Number Theory
    • linear equation
    • Pell equation
    • Thue equation
    • elliptic, hyperelliptic, superelliptic equations
    • general equation, Siegel's finiteness theorem
  2. Baker's method, effective results
  3. From effective to explicit: the Las-Vegas Principle
  4. Continued Fractions and Baker-Davenport Lemma
  5. Explicit solution of simultaneous Pell equations
  6. Explicit solutions of Thue equations.
  7. Elliptic curves, Mordell-Weil Theorem. Solving elliptic equations using elliptic logarithms
Explicit lower bounds for heights of algebraic numbers
Shabnam Akhtari


  1. Heights of Algebraic Numbers
  2. Heights of Vectors and Polynomials
  3. Lehmer's Problem
  4. Effective Lower Bounds for Height of Algebraic Numbers
  5. Effective Computations in Number Fields
Explicit Methods in Algebraic Number Theory
Amalia Pizarro


  1. Introduction: Pell equations and special cases of Fermat last Theorem. Number fields. Ring of integers
  2. Integral bases and discriminant: explicit computations in quadratic and cyclotomic fields.
  3. Unique factorization of ideals in Dedekind domains. Explicit factorization of primes in rings of integers. Ramification.
  4. Ideal class group. Minkowski's bound and finiteness of the class group. Explicit computation in quadratic number fields.
  5. Dirichlet's Theorem. Units in real quadratic fields.
  6. Some computations in PARI/GP.
Lecture 1
Elliptic curves over finite fields with prescribed order
Francesco Pappalardi and Peter Stevenhagen


  1. Introduction to elliptic curves:
    • Weierstrass Equations, the Group Law, the j-Invariant, endomorphisms, division polynomials, the Weil Pairing.
    • Elliptic curves over finite fields, the Frobenius Endomorphism, Hasse-Weil bounds. Schoof's Algorithm for computing the group order, supersingular curves.

  2. Explicit construction of elliptic curves with prescribed order:
    • Complex elliptic curves: Weierstrass parametrization multiplier ring, j-invariant. Complex multiplication. Lattices with given multiplier ring: the class polynomial.
    • Elliptic curves with given point group: reduction mod p, twisting. Explicit construction of cryptographic curves.

    Notes on Complex elliptiuc curves

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