
First week

Finite fields
Michel Waldschmidt (Paris 6)
 Background: group theory, ring theory, field theory, arithmetic. Gauss fields, Cyclotomic polynomials.
 Finite fields: existence, uniqueness, structure, explicit construction. Frobenius automorphisms. Galois Theory of finite fields.

Introduction to Coding Theory
Pierre Arnoux(Luminy)
 Generalities on coding theory.
 Binary codes: Hamming codes, double correcting BCH codes, Melas codes.
 Codes and curves over finite fields.
 Weight distribution of binary codes and number of rational points on curves.

Algebraic curves and the RiemannRoch Theorem
Valerio Talamanca (Roma Tre)
 Review of algebraic prerequisites.
 Weil divisors on algebraic curves: linear equivalence, principal divisors, the Picard group.
 The vector spaces L(D) and their dimension.
 Riemann's theorem: the genus of an algebraic curve.
 Differentials and canonical divisors.
 The theorem of RiemannRoch.

Elliptic curves over finite fields
Francesco Pappalardi(Roma Tre)
 Introduction: Weierstrass Equations, The Group Law, The jInvariant, Elliptic Curves in Characteristic 2, Endomorphisms,
Singular Curves, Elliptic Curves mod n.
 Torsion points: Division Polynomials, The Weil Pairing.
 Elliptic Curves over Finite Fields: The Frobenius Endomorphism, Determining the Group Order, Schoof's Algorithm, Supersingular Curves.

Second week
 Explicit construction of elliptic curves
Peter Stevenhagen (Leiden)
 Complex elliptic curves: Weierstrass parametrization.
 Properties of complex lattices: multiplier ring, jinvariant.
 Elliptic curves over finite fields: Frobenius, HasseWeil bounds.
 Lattices with given multiplier ring: the class polynomial.
 Elliptic curves with given point group: reduction mod p, twisting.
 Explicit construction of cryptographic curves.

Distribution of irreducible polynomials over finite fields
Paul Pollack(University of Georgia)
 Early history (Gauss, Kornblum, Artin)
 Explicit versions of Dirichlet's theorem for polynomials and generalizations, with applications.
 The twin prime conjecture for polynomials and analogues of Hypothesis H (qualitative results).
 The twin prime conjecture for polynomials and analogues of Hypothesis H (quantitative results).
 Irreducible polynomials with restricted coefficients.
 The circle method and sums of irreducible polynomials.

Zeta functions and applications
René Schoof(Tor Vergata)
 Basic properties of zeta functions of algebraic curves over finite fields: Rationality of the zeta function,
Weil polynomial; residue in s=1.
 Examples of the computation of the zeta function of algebraic curves of low genus.
 Riemann Hypothesis for curves (after StepanovBombieri).

Elliptic Curves Criptography
Jorge Jiménez Urroz(UPC Barcelona)
 Generalities on cryptography.
 The Discrete Logarithm Problem: index calculus, attacks with pairings.
 Elliptic Curve Cryptography: the basic setup, DiffieHellman key exchange, MasseyOmura encryption,
ElGamal public key encryption, ElGamal digital Signatures, The Digital Signature Algorithm, ECIES.
 Other applications: Factoring using elliptic curves, primality Testing.

Elliptic curves modulo primes
Alina Cojocaru(University of Illinois at Chicago)
 Distribution of primes.
 Reduction of elliptic curves modulo primes and their groups of rational points.
 LangTrotter conjecture and related topics.
 Average results.
 
The course Computation using PARI and SAGE by Corrado
Falcolini (Roma Tre) will be spread over the two weeks. 
