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A Golod-Shafarevich Equality and p-Tower GroupsMcLeman, Cameron William January 2008 (has links)
Let K be a quadratic imaginary number field, let Kp^(infinity) the top of its p-class field tower for p an odd prime, and let G=Gal(Kp^(infinity)/K). It is known, due to a tremendous collection of work ranging from the principal results of class field theory to the famous Golod-Shafarevich inequality, that G is finite if the p-rank of the class group of K is 0 or 1, and is infinite if this rank is at least 3. This leaves the rank 2 case as the only remaining unsolved case. In this case, while finiteness is still a mystery, much is still known about G: It is a 2-generated, 2-related pro-p-group equipped with an involution that acts as the inverse modulo commutators, and is of one of three possible Zassenhaus types (defined in the paper). If such a group is finite, we will call it an interesting p-tower group. We further the knowledge on such groups by showing that one particular Zassenhaus type can occur as an interesting p-tower group only if the group has order at least p^24 (Proposition 8.1), and by proving a succinct cohomological condition (Proposition 4.7) for a p-tower group to be infinite. More generally, we prove a Golod-Shafarevich equality (Theorem 5.2), refining the famous Golod-Shafarevich inequality, and obtaining as a corollary a strict strengthening of previous Golod-Shafarevich inequalities (Corollary 5.5). Of interest is that this equality applies not only to finite p-groups but also to p-adic analytic pro-p-groups, a class of groups of particular relevance due to their prominent appearance in the Fontaine-Mazur conjecture. This refined version admits as a consequence that the sizes of the first few modular dimension subgroups of an interesting p-tower group G are completely determined by p and its Zassenhaus type, and we compute these sizes. As another application, we prove a new formula (Corollary 5.3) for the Fp-dimensions of the successive quotients of dimension subgroups of free pro-p-groups.
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On towers of function fields over finite fields /Lötter, Ernest C. January 2007 (has links)
Dissertation (PhD)--University of Stellenbosch, 2007. / Bibliography. Also available via the Internet.
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Explicit constructions of asymptotically good towers of function fieldsLotter, Ernest Christiaan 12 1900 (has links)
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.
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On towers of function fields over finite fieldsLotter, Ernest Christiaan 03 1900 (has links)
Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. / Explicit towers of algebraic function fields over finite fields are studied
by considering their ramification behaviour and complete splitting. While
the majority of towers in the literature are recursively defined by a single
defining equation in variable separated form at each step, we consider
towers which may have different defining equations at each step and with
arbitrary defining polynomials.
The ramification and completely splitting loci are analysed by directed
graphs with irreducible polynomials as vertices. Algorithms are exhibited
to construct these graphs in the case of n-step and -finite towers.
These techniques are applied to find new tamely ramified n-step towers
for 1 n 3. Various new tame towers are found, including a family
of towers of cubic extensions for which numerical evidence suggests that
it is asymptotically optimal over the finite field with p2 elements for each
prime p 5. Families of wildly ramified Artin-Schreier towers over small
finite fields which are candidates to be asymptotically good are also considered
using our method.
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