A joint CIMPA-ICTP research school
Algebraic curves over finite fields
University of the Phillipines Dillman, Manila.
July 22nd - August 2nd 2013
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 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.

Second week
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.