On the constant reductions of valued function fields and their automorphism groups

Razafindramahatsiaro, Tovondrainy Christalin (2015-11-20)

Thesis (PhD)--Stellenbosch University, 2015

Thesis

ENGLISH ABSTRACT : The aim of the project is to investigate properties of the automorphism group of a function field in one variable over an algebraically closed field in relation to its reductions with respect to special valuations. Let X be a stable curve defined over a Dedekind scheme S, with smooth generic fiber Xn. It is well known (From Deligne and Mumford) that there exists a natural injective homomorphism between the automorphism groups of Xn and any special fibre of X. In this thesis, we give a generalisation of this theorem in the function field setting of Deuring's theory of constant reductions. The result brings us to one of the central topic in Arithmetic Geometry after Grothendieck, Deligne and Mumford: The lifting problem for curves. We will consider the so-called "weak" Lifting problem for automorphism groups of cyclic curves in this thesis. We will also study good reduction for function fields. In particular, we are interested in corresponding reduction of divisors via the Deuring's arithmetic divisor homomorphism. Together with the generalised Deligne and Mumford Theorem above, we will discuss the Tchebotarev Density Theorem for function fields.

AFRIKAANSE OPSOMMING : Die doel van hierdie projek is om die eienskappe van die Outomorfisme Groep van 'n Funksieliggaam in een veranderlike oor 'n algebraiese afgeslote liggaam in samehang met die reduksies daarvan te studeer. Laat X 'n stabile kurwe wees wat oor 'n Dedekind Scheme S gedefinieerd is met generiese vesel Xn. Dit is bekend, uit werk van Deligne en Mumford, dat daar 'n natuurlike injektiewe homomorfisme tussen die outomorfisme groep van Xn en die van enige spesiale vesel bestaan. In hierdie tesis bewys ons 'n veralgemening van hierdie resultaat in die geval van funksieliggame in die raamwerk van Deuring se teorie van Konstanterekuksie. Die resultaat lei na een van die sentrale onderwerpe in Aritmetiese Meetkunde in die gees van Grothendieck, Deligne en Mumford, naamlik: Die Heffingsprobleem vir kurwes. Ons sal die sogenaamde "Swak Heffingsprobleem"vir die outomorfisme groep van sikliese kurwes in die tesis behandel. Verder bestudeer ons ook vrae binne die raamwerk van die goeie reduksie van kurwes. In besonder stel ons belang in die eienskappe van divisore met behulp van Deuring se divisorreduksie homomorfisme. Deur gebruik te maak van die veralgemening van die Deligne Mumford Stelling wat hierbo na verwys word, bespreek ons die Tschebotarev Digtheidsstelling vir funksieliggame.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/97976
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