Explicit constructions of asymptotically good towers of function fields

[摘要] ENGLISH ABSTRACT: A tower of global function fields :F = (FI, F2' ... ) is an infinite tower of separable extensionsof algebraic function fields of one variable such that the constituent functionfields 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 placesof degree one over the genus of FJWq as i tends to infinity. A tower is called asymptoticallygood if this limit is a positive number. The well-known Drinfeld- Vladutbound provides a general upper bound for this limit.In practise, asymptotically good towers are rare. While the first examples werenon-explicit, we focus on explicit towers of function fields, that is towers where equationsrecursively defining the extensions Fi+d F; are known. It is known that if thefield of constants of the tower has square cardinality, it is possible to attain theDrinfeld- Vladut upper bound for this limit, even in the explicit case. If the field ofconstants does not have square cardinality, it is unknown how close the limit of thetower can come to this upper bound.In this thesis, we will develop the theory required to construct and analyse theasymptotic behaviour of explicit towers of function fields. Various towers will beexhibited, and general families of explicit formulae for which the splitting behaviourand growth of the genus can be computed in a tower will be discussed. When thenecessary theory has been developed, we will focus on the case of towers over fields ofnon-square cardinality and the open problem of how good the asymptotic behaviourof the tower can be under these circumstances.

[发布机构] Stellenbosch University