Department of Mathematical Sciences

Permanent URI for this community

https://scholar.sun.ac.za/handle/10019.1/223

Browse

Browsing Department of Mathematical Sciences by browse.metadata.advisor "Breuer, Florian"

Now showing 1 - 12 of 12
  • Thumbnail Image
    Item
    An analogue of the Andre-Oort conjecture for products of Drinfeld modular surfaces
    (Stellenbosch : Stellenbosch University, 2013-03) Karumbidza, Archie; Breuer, Florian; Keet, A. P.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: This thesis deals with a function eld analog of the André-Oort conjecture. The (classical) André-Oort conjecture concerns the distribution of special points on Shimura varieties. In our case we consider the André-Oort conjecture for special points in the product of Drinfeld modular varieties. We in particular manage to prove the André- Oort conjecture for subvarieties in a product of two Drinfeld modular surfaces under a characteristic assumption.
  • Thumbnail Image
    Item
    Cyclotomic polynomials (in the parallel worlds of number theory)
    (Stellenbosch : Stellenbosch University, 2011-12) Bamunoba, Alex Samuel; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: It is well known that the ring of integers Z and the ring of polynomials A = Fr[T] over a finite field Fr have many properties in common. It is due to these properties that almost all the famous (multiplicative) number theoretic results over Z have analogues over A. In this thesis, we are devoted to utilising this analogy together with the theory of Carlitz modules. We do this to survey and compare the analogues of cyclotomic polynomials, the size of their coefficients and cyclotomic extensions over the rational function field k = Fr(T).
  • Thumbnail Image
    Item
    Drinfeld modular forms of higher rank from a lattice-oriented point of view
    (Stellenbosch : Stellenbosch University., 2020-04) Baker, Liam Bradwin; Basson, Dirk Johannes; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT:
  • Thumbnail Image
    Item
    Drinfeld modules and their application to factor polynomials
    (Stellenbosch : Stellenbosch University, 2012-12) Randrianarisoa, Tovohery Hajatiana; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: Major works done in Function Field Arithmetic show a strong analogy between the ring of integers Z and the ring of polynomials over a nite eld Fq[T]. While an algorithm has been discovered to factor integers using elliptic curves, the discovery of Drinfeld modules, which are analogous to elliptic curves, made it possible to exhibit an algorithm for factorising polynomials in the ring Fq[T]. In this thesis, we introduce the notion of Drinfeld modules, then we demonstrate the analogy between Drinfeld modules and Elliptic curves. Finally, we present an algorithm for factoring polynomials over a nite eld using Drinfeld modules.
  • Thumbnail Image
    Item
    Elliptic curve cryptography
    (Stellenbosch : Stellenbosch University, 2016-12) Louw, Gerard Jacques; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences
    ENGLISH ABSTRACT : In this thesis we present a selection of Diffie-Hellman cryptosystems, which were classically formulated using the multiplicative group of a finite field, but which may be generalised to use other group varieties such as elliptic curves. We also describe known attacks on special cases of such cryptosystems, which manifest as solutions to the discrete logarithm problem for group varieties, and the elliptic curve discrete logarithm problem in particular. We pursue a computational approach throughout, with a focus on the development of practical algorithms.
  • Thumbnail Image
    Item
    Explicit class field theory for rational function fields
    (Stellenbosch : Stellenbosch University, 2008-12) Rakotoniaina, Tahina; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    Class field theory describes the abelian extensions of a given field K in terms of various class groups of K, and can be viewed as one of the great successes of 20th century number theory. However, the main results in class field theory are pure existence results, and do not give explicit constructions of these abelian extensions. Such explicit constructions are possible for a variety of special cases, such as for the field Q of rational numbers, or for quadratic imaginary fields. When K is a global function field, however, there is a completely explicit description of the abelian extensions of K, utilising the theory of sign-normalised Drinfeld modules of rank one. In this thesis we give detailed survey of explicit class field theory for rational function fields over finite fields, and of the fundamental results needed to master this topic.
  • Thumbnail Image
    Item
    Geometric actions of the absolute Galois group
    (Stellenbosch : University of Stellenbosch, 2006-03) Joubert, Paul; Breuer, Florian; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences.
    This thesis gives an introduction to some of the ideas originating from A. Grothendieck's 1984 manuscript Esquisse d'un programme. Most of these ideas are related to a new geometric approach to studying the absolute Galois group over the rationals by considering its action on certain geometric objects such as dessins d'enfants (called stick figures in this thesis) and the fundamental groups of certain moduli spaces of curves. I start by defining stick figures and explaining the connection between these innocent combinatorial objects and the absolute Galois group. I then proceed to give some background on moduli spaces. This involves describing how Teichmuller spaces and mapping class groups can be used to address the problem of counting the possible complex structures on a compact surface. In the last chapter I show how this relates to the absolute Galois group by giving an explicit description of the action of the absolute Galois group on the fundamental group of a particularly simple moduli space. I end by showing how this description was used by Y. Ihara to prove that the absolute Galois group is contained in the Grothendieck-Teichmuller group.
  • Thumbnail Image
    Item
    Geometry of Complex Polynomials: On Sendov's Conjecture
    (Stellenbosch : Stellenbosch University, 2016-12) Chalebgwa, Taboka Prince; Boxall, Gareth John; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences
    ENGLISH ABSTRACT : Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Dégot proved that, for each α ε (0, 1), there is an integer N such that for any polynomial P(z) with degree greater than N, P(a) = 0 and all zeroes inside the unit disk, the disk │z- α│ ≤ 1 contains a critical point of P(z). Basing on this result, we derive an explicit formula N(a) for each α ε (0, 1) and, furthermore, obtain a uniform bound N for all a ε [α,β] where 0 < α < β < 1. This addresses the questions posed in Dégot’s paper.
  • Thumbnail Image
    Item
    On the coefficients of Drinfeld modular forms of higher rank
    (Stellenbosch : Stellenbosch University, 2014-04) Basson, Dirk Johannes; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: Rank 2 Drinfeld modular forms have been studied for more than 30 years, and while it is known that a higher rank theory could be possible, higher rank Drinfeld modular forms have only recently been de ned. In 1988 Gekeler published [Ge2] in which he studies the coe cients of rank 2 Drinfeld modular forms. The goal of this thesis is to perform a similar study of the coe cients of higher rank Drinfeld modular forms. The main results are that the coe cients themselves are (weak) Drinfeld modular forms, a product formula for the discriminant function, the rationality of certain naturally de ned modular forms, and the computation of some Hecke eigenforms and their eigenvalues.
  • Thumbnail Image
    Item
    On the Latimer-MacDuffee theorem for polynomials over finite fields
    (Stellenbosch : University of Stellenbosch, 2011-03) Van Zyl, Jacobus Visser; Breuer, Florian; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: Latimer & MacDuffee showed in 1933 that there is a one-to-one correspondence between equivalence classes of matrices with a given minimum polynomial and equivalence classes of ideals of a certain ring. In the case where the matrices are taken over the integers, Behn and Van der Merwe developed an algorithm in 2002 to produce a representative in each equivalence class. We extend this algorithm to matrices taken over the ring Fq[T] of polynomials over a finite field and prove a modified version of the Latimer-MacDuffee theorem which holds for proper equivalence classes of matrices.
  • Thumbnail Image
    Item
    Riemann hypothesis for the zeta function of a function field over a finite field
    (Stellenbosch : Stellenbosch University, 2013-12) Ranorovelonalohotsy, Marie Brilland Yann; Breuer, Florian; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: See the full text for the abstract
  • Thumbnail Image
    Item
    Torsion bounds for Drinfeld modules with complex multiplication
    (Stellenbosch : Stellenbosch University., 2020-04) Rabenantoandro, Andry Nirina; Breuer, Florian; Wagner, Stephan; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: The main objective of the present thesis is to prove an analogue for Drinfeld modules of a theorem due to Clark and Pollack. The cardinality of the group of K-rational torsion points of an elliptic curve EjK with complex multiplication defined over a number field K of degree d is uniformly bounded by Cd log log d for some absolute and effective constant C > 0, i.e. the constant C > 0 depends neither on E nor on K. Let F be a global function field over Fq and A the ring of elements of F regular away from a fixed prime ¥. Let r 1 be an integer. We prove that there exists a positive constant CA,r > 0 depending only on A and r such that for any field extension L of degree d over F and any Drinfeld A-module jjL of rank r with complex multiplication defined over L and such that the endomorphism ring of j is the maximal order in its CM field, the cardinality of the A-module of L-rational torsion points of j is bounded by CA,rd log log d. The constant depends neither on j nor on L. For a given A and r the constant CA,r is effective and we get an explicit formula for it. The above result is not the full analogue of Clark and Pollack’s theorem but rather a weaker version since it requires the endomorphism ring of j to be the maximal order in its CM field. However, when A = Fq[T], F = Fq(T) and r = 2 we obtain the full analogue of Clark and Pollack’s result by proving the analogue of what they called the Isogeny Torsion Theorem in [CP15].