| 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|
- 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 Riemann-Roch 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 Riemann-Roch.
| Elliptic curves over finite fields
Francesco Pappalardi(Roma Tre)
- Introduction: Weierstrass Equations, The Group Law, The j-Invariant, 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.
| Explicit construction of elliptic curves
Peter Stevenhagen (Leiden)
- Complex elliptic curves: Weierstrass parametrization.
- Properties of complex lattices: multiplier ring, j-invariant.
- Elliptic curves over finite fields: Frobenius, Hasse-Weil 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 Stepanov-Bombieri).
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, Diffie-Hellman key exchange, Massey-Omura 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.
- Lang-Trotter 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. |