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Abstracts of the Conferences
The First International Mathematical School on
Algebraic Geometry in Tunisia
organized by the
Tunisian Women Mathematicians’ Asociation
11-15 September 2017, Hammamet, Tunisia
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Conferences
Mouadh Akriche Real and complex fibrations of low genus. Leila Ben Abdelghani A quick trip through the representation variety of the knot group. Saber Bouanani Li’s criterion and the Riemann hypothesis for function fields. Delphine Boucher Some generalities on the self-dual cyclic codes.
Maher Boudabra Prime numbers - Cryptography - Computational methods over finite fields.
Loubna Ghammam Elliptic curves from useless to indispensable.
Khadija Mbarki Waring's problem, taxicab numbers and other sums of powers.
Mohammad Sadek Elliptic curves and continued fractions.
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Real and complex fibrations of low genus
Mouadh AKRICHE
Institut Préparatoire aux Études d’Ingénieur de Bizerte, Tunisia
Abstract
The aim of this talk is to give some new contributions to the study of the
topology and the geometry of real and complex fibrations of low genus.
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A quick trip through the representation variety of a knot group
Leila BEN ABDELGHANI
Faculty of Sciences of Monastir, Tunisia
Abstract
Research has shown that many topological properties of a knot K in are encoded in the
representation and character varieties of its group, so it is of interest to study their basic
properties. We denote by R(π):=Hom(π, ) the representation variety of a knot group
π. A representation is called irreducible if the only subspaces of which
are invariant under are {0} and By a result of Thurston, under certain conditions,
one can deform an irreducible representation non trivially but there is no general theorem
which allows the deformation of reducible representations. The aim of this talk is to give an
idea about the local structure of the variety of representations π π
and more generally of Hom(π, ), in the neighborhood of an abelian or a reducible
representation.
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Li's criterion and the Riemann hypothesis for function fields
Saber BOUANANI
Faculty of Sciences of Tunis, Tunisia
Abstract
We extend Li's criterion for a function field of genus g over a finite field . We prove
that the zeros of the zeta-function of lie on the line
if and only if the Li
coefficients satisfy
for all . Therefore, we particularly show
that the Riemann hypothesis for the function field holds if and only if
for all , where ( )| is the number of -rational points on the curve
associated to the function field . We give an explicit asymptotic formula for the Li
coefficients Finally, we show that the Li coefficients are increasing in .
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Some generalities on the self-dual cyclic codes
Delphine BOUCHER
Université de Rennes 1, France
Abstract
Self-dual codes and cyclic codes have been extensively studied on finite fields. The cyclic
codes of length n over a finite field F can be completely determined by their generator
polynomials. These are the divisors of x^n-1 in F[x]. The polynomials generating the self-
dual cyclic codes are the solutions of a polynomial equation in F[x]. It can be solved using
factorization properties of x^n-1 (Sloane and Thompson, 1983). We can define "skew cyclic
codes" by replacing the ring F[x] with the ring of the skew polynomials F [x; theta] where
theta is a non trivial automorphism of F (B., Geiselmann, Ulmer, 2007). In this ring,
multiplication is defined by the non-commutative rule "x. a = theta(a) x" for all a in F. We
can prove that "skew cyclic codes" are characterized by an equation called "self-dual skew
equation ". The purpose of this talk is to describe the construction of Sloane and Thomson
of self-dual cyclic codes and to show how to adapt this method to solve the "self-dual skew
equation" in some particular cases.
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Elliptic Curves : KMOV Cryptosystem
Maher BOUDABRA
Université de Caen (France)
Abstract
The KMOV public key cryptosystem is a public key cryptosystem based on
elliptic curves modulo an RSA modulus n = pq. KMOV is more resistant than the
RSA cryptosystem to the attacks that are not based on factorization. In this
lecture, we propose a generalization of the KMOV cryptosystem with a prime
power modulus of the form n = p^rq^s. We discuss its efficiency and its
resistance to non factorization attacks.
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Elliptic curves from useless to the indispensable
Loubna GHAMMAM
Université Rennes 1, France
Abstract
The pairings on elliptic curves are mathematical tools introduced by André Weil in 1948.
These are a very popular subjects over the last years on asymmetric cryptography. They
allow to perform some cryptographic operations which are not performed easily without
pairings as for the short signature and the identity based cryptography. In recent years, the
calculation of pairings has become easier thanks to the introduction of new mathematical
optimisations on elliptic curves which are called the pairing-friendly ellipic curves. Next
step is to transfer this technology from the theory to the practical implementation which is
executed on some electronic components as FPGA. In this talk, I will introduce at first the
elliptic curves, then the pairings over these curves and I will ended my exposee by
presenting the problem of the pairing implementation in a resricted environnements.
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Waring's problem, taxicab numbers and other sums of powers
Khadija MBARKI
Faculty of Sciences of Monastir, Tunisia
Abstract In this talk, I will begin by considering the most ancient of number theory problems, that of
expressing a square as a sum of two squares. I will give some results on which numbers can
be expressed as a sum of two squares in various numbers of ways, using some elementary
results from the theory of quadratic forms. And I will show which numbers can be
expressed as sums of three squares (most, due to Gauss) and four squares (all, due to
Lagrange). I will then examine the problem that made the number 1729 famous, which
numbers are sums of two cubes in two different ways? I will present a probabilistic
approach to predicting the number of solutions to this problem. I will give bounds on the
number of ways of writing a number as a sum of two cubes, both based on its size and on
its factorization. And I will present parameterizations that gives infinite families of
solutions. Finally, I will address what is usually known as ”Waring’s problem”-how many nth
powers are needed to write any number as a sum of such powers? We will see the basic
results for small powers of positive integers, including the variation of this problem known
as the ”easier” Waring’s problem in which sums and differences of powers are taken.
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Elliptic Curves and Continued Fractions
Mohammad SADEK
The American University in Cairo, Egypt
Abstract
Given a positive rational number N, the infinite continued fraction expansion of
is periodic. Unfortunately, if is a monic polynomial of even degree
with rational coefficients, then the continued fraction expansion of is
not necessarily periodic. One may ask which of these polynomials enjoy this
periodicity property. We display how elliptic curves give insight into the
question.