Professor Florian Breuer

Professor Florian Breuer

Professor

School of Mathematical and Physical Sciences (Mathematics)

Career Summary

Biography

Research Interests: Florian works in Number Theory, especially in elliptic curves, Drinfeld modules, Drinfeld modular forms and also elementary topics such as Ducci sequences.

Biography: Florian grew up in Stellenbosch, South Africa, where he completed school as well as his undergraduate studies in Mathematics and Theoretical Physics. He then obtained a bursary from the French government to complete his graduate studies in Paris, where he completed his D.E.A. (Masters) at the Université Pierre et Marie Curie (Paris 6) in 1999 and his PhD at the Université Denis Diderot (Paris 7) in 2002, both under supervision of Marc Hindry.

After two postdoctoral fellowships in Taiwan and Germany, he returned to Stellenbosch University in July 2004 as a senior lecturer. He was promoted to associate professor in 2007 and spent a semester in Germany and Switzerland in 2009 on an Alexander-von-Humboldt Fellowship for Experienced Researchers. Florian served as head of the Mathematics Division at Stellenbosch University from 2012 to 2015, and was promoted to full professor in 2013. 

In April 2018 Florian and his family moved to the University of Newcastle. He still holds an appointment as extraordinary professor at Stellenbosch University.

Florian is on the executive of the Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA) and serves on the editorial boards of the Journal of Number Theory and Quaestiones Mathematicae.

Reseach: In modern Number Theory, there exist the parallel worlds of number fields and function fields. Most of the well-known results and phenomena of Number Theory, such as multiplicative Number Theory, the Riemann Hypothesis, Class Field Theory and modular forms have parallels in the function field world. In particular, Drinfeld modules are function field objects whose behaviour closely parallels that of elliptic curves over number fields.

Florian's main research area is the arithmetic of function fields. This started with his PhD thesis, in which he proved an analogue of the André-Oort Conjecture for products of Drinfeld modular curves. He later extended this conjecture to subvarieties of Drinfeld modular varieties, and proved it in certain special cases. From here his research voyage has lead him in a natural way to consider Drinfeld modular polynomials in higher rank, Galois representations associated to Drinfeld modules (which allowed him to prove the Generalised Iteration Conjecture of Abhyankar) and to Drinfeld modular forms. Most recently he was involved in joint work with former PhD student Dirk Basson (Stellenbosch), and friend and mentor Richard Pink (ETH-Zurich), which laid the foundations for the analytic theory of Drinfeld modular forms in arbitrary rank. This research has opened the door to a variety of exciting new directions.

In other work, Florian is interested in elliptic curves, and has contributed results on the growth of torsion subgroups of elliptic curves over number fields and has studied Heegner points on elliptic curves. 

A more elementary, but rich and interesting, topic is Ducci sequences. Florian's first research experience as an undergraduate student concerned periods of Ducci sequences, and he keeps returning to this topic over the years.


Qualifications

  • Doctoral Degree in Mathematics (Equiv PhD), University of Paris - France

Keywords

  • Drinfeld modular forms
  • Drinfeld modules
  • Ducci sequences
  • Elliptic curves
  • Number Theory

Languages

  • German (Mother)
  • English (Fluent)
  • French (Fluent)
  • Afrikaans (Fluent)

Fields of Research

Code Description Percentage
010102 Algebraic and Differential Geometry 10
010101 Algebra and Number Theory 90

Professional Experience

UON Appointment

Title Organisation / Department
Professor University of Newcastle
School of Mathematical and Physical Sciences
Australia

Academic appointment

Dates Title Organisation / Department
1/04/2018 - 31/03/2021 Extraordinary Professor Stellenbosch University
Mathematical Sciences
South Africa
1/01/2013 - 31/03/2018 Professor Stellenbosch University
Mathematical Sciences
South Africa
1/10/2007 - 31/12/2012 Associate Professor Stellenbosch University
Mathematical Sciences
South Africa
1/07/2004 - 30/09/2007 Senior Lecturer Stellenbosch University
Mathematical Sciences
South Africa
1/01/2004 - 30/04/2004 Postdoctoral Fellow Max Planck Institute
Max-Planck-Institute for Mathematics
Germany
1/02/2003 - 31/10/2003 Postdoctoral Fellow National Tsing Hua University
National Center for Theoretical Sciences
Taiwan, Province of China

Awards

Prize

Year Award
2008 Meiring-Naude Medal
The Royal Society of South Africa

Scholarship

Year Award
2009 Alexander-von-Humboldt Fellowship for Experienced Researchers
Alexander Von Humboldt Foundation

Teaching

Code Course Role Duration
1210 Mathematical Discovery 1
The University of Newcastle
Professor 1/04/2018 - 30/06/2018
3120 Algebra
The University of Newcastle
Lectuer, course coordinator 30/07/2018 - 30/11/2018
4104 Number Theory
The University of Newcastle
Lecturer, course coordinator 30/07/2018 - 30/11/2018
1120 Mathematics for Engineering, Science and Technology, 2
The University of Newcastle
Lecturer, course coordinator 25/02/2019 - 29/06/2019
1210 Mathematical Discovery 1
The University of Newcastle
Lecturer, course coordinator 25/02/2019 - 29/06/2019
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Publications

For publications that are currently unpublished or in-press, details are shown in italics.


Chapter (1 outputs)

Year Citation Altmetrics Link
2005 Breuer F, Pink R, 'Monodromy groups associated to non-isotrivial Drinfeld modules in generic characteristic', , BIRKHAUSER BOSTON 61-69 (2005)
Citations Scopus - 4Web of Science - 3

Journal article (28 outputs)

Year Citation Altmetrics Link
2019 Breuer F, 'PERIODS of DUCCI SEQUENCES and ODD SOLUTIONS to A PELLIAN EQUATION', Bulletin of the Australian Mathematical Society, (2019)

© 2019 Australian Mathematical Publishing Association Inc. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creative... [more]

© 2019 Australian Mathematical Publishing Association Inc. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. A Ducci sequence is a sequence of integer -tuples generated by iterating the map Such a sequence is eventually periodic and we denote by the maximal period of such sequences for given. We prove a new upper bound in the case where is a power of a prime for which is a primitive root and the Pellian equation has no solutions in odd integers and.

DOI 10.1017/S0004972719000212
2017 Basson D, Breuer F, 'On certain Drinfeld modular forms of higher rank', Journal de Théorie des Nombres de Bordeaux, 29 827-843 (2017)
Citations Scopus - 4Web of Science - 2
2016 Breuer F, 'A note on Gekeler's h-function', ARCHIV DER MATHEMATIK, 107 305-313 (2016)
DOI 10.1007/s00013-016-0923-1
Citations Scopus - 1Web of Science - 1
2016 Breuer F, 'Explicit Drinfeld moduli schemes and Abhyankar's Generalized Iteration Conjecture', JOURNAL OF NUMBER THEORY, 160 432-450 (2016)
DOI 10.1016/j.jnt.2015.08.021
Citations Scopus - 1Web of Science - 1
2016 Breuer F, Rueck H-G, 'Drinfeld modular polynomials in higher rank II: Kronecker congruences', JOURNAL OF NUMBER THEORY, 165 1-14 (2016)
DOI 10.1016/j.jnt.2016.01.001
2012 Breuer F, 'Newton Identities for Weierstrass Products', AMERICAN MATHEMATICAL MONTHLY, 119 796-799 (2012)
DOI 10.4169/amer.math.monthly.119.09.796
2012 Breuer F, 'Special subvarieties of Drinfeld modular varieties', JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 668 35-57 (2012)
DOI 10.1515/CRELLE.2011.136
Citations Scopus - 2Web of Science - 3
2010 Breuer F, 'Torsion bounds for elliptic curves and Drinfeld modules', JOURNAL OF NUMBER THEORY, 130 1241-1250 (2010)
DOI 10.1016/j.jnt.2009.11.009
Citations Scopus - 10Web of Science - 10
2010 Breuer F, 'Ducci sequences and cyclotomic fields', JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, 16 847-862 (2010)
DOI 10.1080/10236190802566509
Citations Scopus - 4Web of Science - 4
2009 Breuer F, Rueck H-G, 'Drinfeld modular polynomials in higher rank', JOURNAL OF NUMBER THEORY, 129 59-83 (2009)
DOI 10.1016/j.jnt.2008.07.010
Citations Scopus - 3Web of Science - 3
2008 Breuer F, Im B-H, 'Heegner points and the rank of elliptic curves over large extensions of global fields', CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 60 481-490 (2008)
DOI 10.4153/CJM-2008-023-0
Citations Scopus - 2Web of Science - 2
2007 Breuer F, 'Ducci sequences in higher dimensions', Integers: Electronic Journal of Combinatorial Number Theory, 7 A24-A24 (2007)
2007 Breuer F, 'CM points on products of Drinfeld modular curves', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 359 1351-1374 (2007)
DOI 10.1090/S0002-9947-06-04109-2
Citations Scopus - 2Web of Science - 2
2007 Breuer F, Lotter E, van der Merwe B, 'Ducci-sequences and cyclotomic polynomials', FINITE FIELDS AND THEIR APPLICATIONS, 13 293-304 (2007)
DOI 10.1016/j.ffa.2005.11.003
Citations Scopus - 8Web of Science - 7
2007 Breuer F, 'The Andre-Oort conjecture for Drinfeld modular varieties', COMPTES RENDUS MATHEMATIQUE, 344 733-736 (2007)
DOI 10.1016/j.crma.2007.05.008
2005 Breuer F, 'The Andre-Oort conjecture for products of Drinfeld modular curves', JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 579 115-144 (2005)
Citations Scopus - 5Web of Science - 5
2004 Breuer F, 'Images of isogeny classes on modular elliptic curves', MATHEMATICAL RESEARCH LETTERS, 11 649-651 (2004)
Citations Web of Science - 1
2004 Breuer F, 'Higher Heegner points on elliptic curves over function fields', JOURNAL OF NUMBER THEORY, 104 315-326 (2004)
DOI 10.1016/j.jnt.2003.09.001
Citations Scopus - 7Web of Science - 7
2002 Breuer F, 'La conjecture d André Oort pour le produit de deux courbes modulaires de Drinfeld', Comptes Rendus Mathematique, 335 867-870 (2002)
2002 Breuer F, 'Distinguished liftings and the andré-oort conjecture', Quaestiones Mathematicae, 25 363-380 (2002)

In this paper we study liftings of affine varieties from finite fields to number fields, such that the lifted varieties contain specified ¿canonical¿ lifts of points. If this cano... [more]

In this paper we study liftings of affine varieties from finite fields to number fields, such that the lifted varieties contain specified ¿canonical¿ lifts of points. If this canonical lifting of points corresponds to the Deuring-Serre-Tate lift of j-invariants of ordinary elliptic curves, then the resulting lifting problem is closely related to the André-Oort conjecture. We explore this connection, prove some results related to the André-Oort conjecture, and then apply these results together with other known special cases of the conjecture to our lifting problems. © 2002, Taylor & Francis Group, LLC. All rights reserved.

DOI 10.2989/16073600209486023
2002 Breuer F, 'The André-Oort conjecture for the product of two Drinfeld modular curves', Comptes Rendus Mathematique, 335 867-870 (2002)

We prove an analogue of the André-Oort conjecture for the product of two Drinfeld modular curves, following S.J. Edixhoven's approach. © 2002 Académie des sciences/Éditions s... [more]

We prove an analogue of the André-Oort conjecture for the product of two Drinfeld modular curves, following S.J. Edixhoven's approach. © 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.

DOI 10.1016/S1631-073X(02)02589-X
Citations Scopus - 2Web of Science - 2
2001 Breuer F, 'Heights of CM points on complex affine curves', Ramanujan Journal, 5 311-317 (2001)

In this note we show that, assuming the generalized Riemann hypothesis for quadratic imaginary fields, an irreducible algebraic curve in C'' is modular if and only if it... [more]

In this note we show that, assuming the generalized Riemann hypothesis for quadratic imaginary fields, an irreducible algebraic curve in C'' is modular if and only if it contains a CM point of sufficiently large height. This is an effective version of a theorem of Edixhoven.

DOI 10.1023/A:1012982812988
Citations Scopus - 12Web of Science - 10
1999 Breuer F, 'Ducci sequences over abelian groups', COMMUNICATIONS IN ALGEBRA, 27 5999-6013 (1999)
DOI 10.1080/00927879908826804
Citations Scopus - 4Web of Science - 5
1998 Breuer F, 'A note on a paper by Glaser and Schoffl', FIBONACCI QUARTERLY, 36 463-466 (1998)
Citations Scopus - 6Web of Science - 6
1998 Breuer F, Robson JM, 'Strategy and complexity of the game of squares', BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 30 274-282 (1998)
DOI 10.1112/S0024609397004001
1997 Poretti E, Koen C, Martinez P, Breuer F, De Alwis D, Haupt H, 'Discovery and analysis of Gamma Doradus type pulsations in the F0 IV star HR 2740= QW PUP', Monthly Notices of the Royal Astronomical Society, 292 621-630 (1997)
Citations Scopus - 10Web of Science - 13
1997 Poretti E, Koen C, Martinez P, Breuer F, de Alwis D, Haupt H, 'Discovery and analysis of Gamma Doradus type pulsations in the Cloud and the effect of metallicity on pulsation', arXiv preprint astro-ph/9706212, (1997)
1997 Breuer F, 'A NOTE ON A PAPER BY GLASER ANB SCHOFFL (1997)
Show 25 more journal articles

Conference (2 outputs)

Year Citation Altmetrics Link
2012 Breuer F, 'On Abhyankar s Generalized Iteration Conjecture', DMV-Jahrestagung 2012 (2012)
1998 Poretti E, Mantegazza L, Koen C, Martinez P, Breuer F, De Alwis D, Haupt H, 'Line Profile Variations in the Spectra of the Dor Star HR 2740', Symposium-International Astronomical Union, Cambridge University Press (1998)

Other (1 outputs)

Year Citation Altmetrics Link
2013 Breuer F, 'The parallel worlds of number theory', (2013) [O1]

Thesis / Dissertation (1 outputs)

Year Citation Altmetrics Link
2002 Breuer F, Sur la conjecture d André-Oort et courbes modulaires de Drinfeld, (2002)
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Grants and Funding

Summary

Number of grants 15
Total funding $273,720

Click on a grant title below to expand the full details for that specific grant.


20191 grants / $15,320

Alexander-von-Humboldt Renewed Research Stay$15,320

Arithmetic and analysis on Drinfeld moduli schemes

Funding body: Alexander Von Humboldt Foundation

Funding body Alexander Von Humboldt Foundation
Project Team

Florian Breuer, Gebhard Böckle

Scheme Alexander-von-Humboldt Fellowship for Experienced Researchers
Role Lead
Funding Start 2019
Funding Finish 2019
GNo
Type Of Funding International - Competitive
Category 3IFA
UON N

20182 grants / $55,000

Startup Grant$50,000

Funding body: The University of Newcastle

Funding body The University of Newcastle
Project Team

Florian Breuer

Scheme School of Mathematical and Physical Sciences
Role Lead
Funding Start 2018
Funding Finish 2019
GNo
Type Of Funding Internal
Category INTE
UON N

AMSI/AustMS Workshop on Mathematical Thinking$5,000

Funding for the AMSI/AustMS Workshop on Mathematical Thinking, held in Newcastle, 14-16 November 2018

Funding body: AMSI Intern Australian Mathematical and Physical Sciences

Funding body AMSI Intern Australian Mathematical and Physical Sciences
Project Team

Florian Breuer, Ljiljana Brankovic, Judy-anne Osborn, Tomothy Trudgian

Scheme Small Event Funding
Role Lead
Funding Start 2018
Funding Finish 2018
GNo
Type Of Funding Aust Competitive - Commonwealth
Category 1CS
UON N

20172 grants / $25,300

Stellenbosch-AIMS Number Theory Conference$21,300

Funding for international Number Theory conference

Funding body: CoE-MASS

Funding body CoE-MASS
Project Team

Florian Breuer, Barry Green, Patrick Rabarison

Scheme Number Theory
Role Lead
Funding Start 2017
Funding Finish 2017
GNo
Type Of Funding External
Category EXTE
UON N

Africa Collaboration Grant - conference funding$4,000

Funding body: Stellenbosch University

Funding body Stellenbosch University
Project Team

Florian Breuer, Barry Green, Patrick Rabarison

Scheme Africa Collaboration Grant
Role Lead
Funding Start 2017
Funding Finish 2017
GNo
Type Of Funding Internal
Category INTE
UON N

20161 grants / $39,000

Drinfeld modular forms and L-Functions$39,000

Funding body: University of Stellenbosch

Funding body University of Stellenbosch
Project Team

Florian Breuer, Luca Demangos

Scheme Subcommittee-B Postdoc Grant
Role Lead
Funding Start 2016
Funding Finish 2018
GNo
Type Of Funding Internal
Category INTE
UON N

20111 grants / $2,000

Africa Collaboration Grant - Exchange with Madagascar$2,000

Funding body: Stellenbosch University

Funding body Stellenbosch University
Project Team

Florian Breuer, David Holgate, Stephan Wagner

Scheme Africa Collaboration Grant
Role Lead
Funding Start 2011
Funding Finish 2011
GNo
Type Of Funding Internal
Category INTE
UON N

20094 grants / $106,500

Incentive Funding for Rated Researchers$38,500

Yearly funding for rated researchers

Funding body: National Research Foundation

Funding body National Research Foundation
Project Team

Florian Breuer

Scheme Incentive funding for rated researchers
Role Lead
Funding Start 2009
Funding Finish 2018
GNo
Type Of Funding External
Category EXTE
UON N

Drinfeld modular forms of higher rank$30,300

Funding body: Alexander Von Humboldt Foundation

Funding body Alexander Von Humboldt Foundation
Project Team

Florian Breuer, Hans-Georg Rueck

Scheme Alexander-von-Humboldt Fellowship for Experienced Researchers
Role Lead
Funding Start 2009
Funding Finish 2009
GNo
Type Of Funding International - Competitive
Category 3IFA
UON N

Drinfeld modular forms in higher rank$29,700

Funding body: National Research Foundation

Funding body National Research Foundation
Project Team

Florian Breuer

Scheme Blue Skies Research Grant
Role Lead
Funding Start 2009
Funding Finish 2011
GNo
Type Of Funding External
Category EXTE
UON N

ALGANT Mobility Grant$8,000

Funding body: ALGANT

Funding body ALGANT
Project Team

Florian Breuer

Scheme ALGANT mobility grant
Role Lead
Funding Start 2009
Funding Finish 2009
GNo
Type Of Funding External
Category EXTE
UON N

20061 grants / $1,500

IMU Travel Grant$1,500

Travel grant to attend 2006 ICM in Madrid

Funding body: International Mathematical Union

Funding body International Mathematical Union
Project Team

Florian Breuer

Scheme IMU Travel Grant
Role Lead
Funding Start 2006
Funding Finish 2006
GNo
Type Of Funding International - Competitive
Category 3IFA
UON N

20052 grants / $24,000

Arithmetic Geometry$14,000

Funding body: Stellenbosch University

Funding body Stellenbosch University
Project Team

Florian Breuer

Scheme Subcommittee-B Fund for Promising Young Researchers
Role Lead
Funding Start 2005
Funding Finish 2007
GNo
Type Of Funding Internal
Category INTE
UON N

Arithmetic Geometry$10,000

Funding body: Stellenbosch University

Funding body Stellenbosch University
Project Team

Florian Breuer

Scheme Subcommittee-B Research Grant
Role Lead
Funding Start 2005
Funding Finish 2006
GNo
Type Of Funding Internal
Category INTE
UON N

20041 grants / $5,100

Conference travel grants$5,100

Funding body: Stellenbosch University

Funding body Stellenbosch University
Project Team

Florian Breuer

Scheme Faculty of Science
Role Lead
Funding Start 2004
Funding Finish 2008
GNo
Type Of Funding Internal
Category INTE
UON N
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Research Supervision

Number of supervisions

Completed12
Current3

Current Supervision

Commenced Level of Study Research Title Program Supervisor Type
2018 PhD On Drinfeld modular forms Mathematics, Stellenbosch University Co-Supervisor
2011 PhD Torsion bounds for CM Drinfeld modules Mathematics, Stellenbosch University Sole Supervisor
2011 PhD Bounds on coefficients of Drinfeld modular polynomials Mathematics, University of Antananarivo Principal Supervisor

Past Supervision

Year Level of Study Research Title Program Supervisor Type
2018 Post-Doctoral Fellowship Quantum j-invariants and class fields of function fields Mathematics, Stellenbosch University Sole Supervisor
2016 Masters Geometry of Complex Polynomials: On Sendov’s Conjecture
Sendov&rsquo;s conjecture states that if all the zeroes of a complex polynomial<br />P(z) of degree at least two lie in the unit disk, then within a unit distance<br />of each zero lies a critical point of P(z). In a paper that appeared in 2014,<br />D&eacute;got proved that, for each a in (0, 1), there is an integer N such that for any<br />polynomial P(z) with degree greater than N, P(a) = 0 and all zeroes inside<br />the unit disk, the disk |z - a| &lt;= 1 contains a critical point of P(z). Basing<br />on this result, we derive an explicit formula N(a) for each a in (0, 1) and,<br />furthermore, obtain a uniform bound N for all a in [alpha, beta] where 0 &lt; alpha &lt; beta &lt;<br />1. This addresses the questions posed in D&eacute;got&rsquo;s paper.<br />
Mathematics, Stellenbosch University Principal Supervisor
2016 Masters Elliptic Curve Cryptography
In this thesis we present a selection of Diffie-Hellman cryptosystems, which<br />were classically formulated using the multiplicative group of a finite field, but<br />which may be generalised to use other group varieties such as elliptic curves.<br />We also describe known attacks on special cases of such cryptosystems, which<br />manifest as solutions to the discrete logarithm problem for group varieties,<br />and the elliptic curve discrete logarithm problem in particular. We pursue<br />a computational approach throughout, with a focus on the development of<br />practical algorithms.
Mathematics, Stellenbosch University Sole Supervisor
2015 Post-Doctoral Fellowship Drinfeld modular forms in higher rank Mathematics, Stellenbosch University Sole Supervisor
2013 PhD On the coefficients of Drinfeld modular forms of higher rank
<p><span lang="EN-GB">While defined on a different number system than the usual real numbers, Drinfeld modular forms are functions which exhibit remarkable symmetry properties. The 1-dimensional Drinfeld modular forms are well understood and correspond closely to classical modular forms which have a central position in the solutions of many important problems in modern mathematics. Recently, higher dimensional Drinfeld modular forms have been defined, but not much is known about them at present. The candidate has made important progress toward the understanding of these functions.</span></p>
Mathematics, Stellenbosch University Sole Supervisor
2013 Masters Riemann Hypothesis for the zeta function of a function field over a finite field.
Let K be a function field over a finite field. Fix a place (\infty) of K, which<br />we shall call the prime at infinity. We consider the ring A of elements of K regular away from infinity,&nbsp;which we call the ring of integers<br />of K with respect to (\infty). There is a bijection between the set of proper ideals<br />of A and the places of K different from (\infty). We define the zeta function Z_A(s)<br />for the ring A in a way analogous to the Dedekind zeta function of the ring of<br />integers of a number field. The analogue of the Riemann Hypothesis for Z_A(s)<br />was first proved by Andr&eacute; Weil in 1948, and our goal is to give an exposition<br />of a simpler proof of this theorem due to Enrico Bombieri.
Mathematics, Stellenbosch University Sole Supervisor
2012 Masters Drinfeld modules and their application to polynomial factorization
<p><span style="font-family:Arial, Helvetica, sans-serif;">Major works done in Function Field Arithmetic show strong analogy between </span></p><p><span style="font-family:Arial, Helvetica, sans-serif;">the ring of integers Z and ring of polynomials over a finite field F [T]. While</span></p><p><span style="font-family:Arial, Helvetica, sans-serif;">an algorithm has been discovered to factor integers using elliptic curves, the</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">discovery of Drinfeld modules, which are analogous to elliptic curves made it</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">possible to exhibit an algorithm for factorising polynomials in the ring F [T].</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">In this thesis, we will introduce the notion of Drinfeld modules by studying</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">some notion within it. Then we will show an evidence of the analogy between</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">Drinfeld modules and Elliptic curves. Finally, we will confirm the analogy by</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">giving the algorithm for factoring polynomials over finite field using Drinfeld</span><br /><span style="font-family:Arial, Helvetica, sans-serif;">modules.</span></p>
Mathematics, Stellenbosch University Sole Supervisor
2012 PhD An analogue of the André-Oort conje