Browsing by Author "Razafindramahatsiaro, Tovondrainy Christalin"

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    On the constant reductions of valued function fields and their automorphism groups
    (Stellenbosch : Stellenbosch University, 2015-11-20) Razafindramahatsiaro, Tovondrainy Christalin; Green, Barry W.; Stellenbosch University. Faculty of Science. Department Mathematical Sciences (Mathematics)
    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.