Thesis (MSc)--Stellenbosch University, 2003 / ENGLISH ABSTRACT: A tower of global function fields :F = (FI, F2' ... ) is an infinite tower of separable extensions
of algebraic function fields of one variable such that the constituent function
fields have the same (finite) field of constants and the genus of these tend to infinity.
A study can be made of the asymptotic behaviour of the ratio of the number of places
of degree one over the genus of FJWq as i tends to infinity. A tower is called asymptotically
good if this limit is a positive number. The well-known Drinfeld- Vladut
bound provides a general upper bound for this limit.
In practise, asymptotically good towers are rare. While the first examples were
non-explicit, we focus on explicit towers of function fields, that is towers where equations
recursively defining the extensions Fi+d F; are known. It is known that if the
field of constants of the tower has square cardinality, it is possible to attain the
Drinfeld- Vladut upper bound for this limit, even in the explicit case. If the field of
constants does not have square cardinality, it is unknown how close the limit of the
tower can come to this upper bound.
In this thesis, we will develop the theory required to construct and analyse the
asymptotic behaviour of explicit towers of function fields. Various towers will be
exhibited, and general families of explicit formulae for which the splitting behaviour
and growth of the genus can be computed in a tower will be discussed. When the
necessary theory has been developed, we will focus on the case of towers over fields of
non-square cardinality and the open problem of how good the asymptotic behaviour
of the tower can be under these circumstances. / AFRIKAANSE OPSOMMING: 'n Toring van globale funksieliggame F = (FI, F2' ... ) is 'n oneindige toring van
skeibare uitbreidings van algebraïese funksieliggame van een veranderlike sodat die
samestellende funksieliggame dieselfde (eindige) konstante liggaam het en die genus
streef na oneindig. 'n Studie kan gemaak word van die asimptotiese gedrag van die
verhouding van die aantal plekke van graad een gedeel deur die genus van Fi/F q soos
i streef na oneindig. 'n Toring word asimptoties goed genoem as hierdie limiet 'n
positiewe getal is. Die bekende Drinfeld- Vladut grens verskaf 'n algemene bogrens
vir hierdie limiet.
In praktyk is asimptoties goeie torings skaars. Terwyl die eerste voorbeelde nie
eksplisiet was nie, fokus ons op eksplisiete torings, dit is torings waar die vergelykings
wat rekursief die uitbreidings Fi+d F; bepaal bekend is. Dit is bekend dat as
die kardinaliteit van die konstante liggaam van die toring 'n volkome vierkant is, dit
moontlik is om die Drinfeld- Vladut bogrens vir die limiet te behaal, selfs in die eksplisiete
geval. As die konstante liggaam nie 'n kwadratiese kardinaliteit het nie, is
dit onbekend hoe naby die limiet van die toring aan hierdie bogrens kan kom.
In hierdie tesis salons die teorie ontwikkel wat benodig word om eksplisiete torings
van funksieliggame te konstrueer, en hulle asimptotiese gedrag te analiseer. Verskeie
torings sal aangebied word en algemene families van eksplisiete formules waarvoor die
splitsingsgedrag en groei van die genus in 'n toring bereken kan word, sal bespreek
word. Wanneer die nodige teorie ontwikkel is, salons fokus op die geval van torings
oor liggame waarvan die kardinaliteit nie 'n volkome vierkant is nie, en op die oop
probleem aangaande hoe goed die asimptotiese gedrag van 'n toring onder hierdie
omstandighede kan wees.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/53417 |
| Date | 12 1900 |
| Creators | Lotter, Ernest Christiaan |
| Contributors | Green, B. W., Van der Merwe, A. B., Stellenbosch University. Faculty of Science. Department of Mathematical Sciences. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
| Format | vi, 106 pages : illustrations |
| Rights | Stellenbosch University |
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