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 Chebyshev estimates; Abel summation formula; The Moebius function; The prime number theorem; Mean values of arithmetic functions; The Brun pure sieve. Introduction to algebraic number theory Valerio Talamanca Number fields Norms, traces, and discriminants Ring of integers Ideal factorization in Dedekind rings Decomposition and ramification 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 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 Baker's method, effective results From effective to explicit: the Las-Vegas Principle Continued Fractions and Baker-Davenport Lemma Explicit solution of simultaneous Pell equations Explicit solutions of Thue equations. Elliptic curves, Mordell-Weil Theorem. Solving elliptic equations using elliptic logarithms Explicit lower bounds for heights of algebraic numbersShabnam Akhtari Heights of Algebraic Numbers Heights of Vectors and Polynomials Lehmer's Problem Effective Lower Bounds for Height of Algebraic Numbers Effective Computations in Number Fields
 Explicit Methods in Algebraic Number Theory Amalia Pizarro Introduction: Pell equations and special cases of Fermat last Theorem. Number fields. Ring of integers Integral bases and discriminant: explicit computations in quadratic and cyclotomic fields. Unique factorization of ideals in Dedekind domains. Explicit factorization of primes in rings of integers. Ramification. Ideal class group. Minkowski's bound and finiteness of the class group. Explicit computation in quadratic number fields. Dirichlet's Theorem. Units in real quadratic fields. Some computations in PARI/GP. Lecture 1,,Lecture 2 Lecture 3, Lecture 4-5 Elliptic curves over finite fields with prescribed order Francesco Pappalardi and Peter Stevenhagen 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. 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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