@MISC{Uzunkol_researchstatement, author = {Osmanbey Uzunkol}, title = {Research Statement}, year = {} }

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Abstract

My research is mainly in algorithmic number theory and arithmetic geometry with particular interest in cryptography, coding theory, complex multiplication theory and explicit class field theory. This outline is intended as a brief description of the research projects I have undertaken. It projects forward to research projects I am currently working on and others I am planning to undertake after my Phd. 1 Construction of elliptic and hyperelliptic curves over finite fields with CM method Since Kronecker, number theorists have kept exploiting the idea of generating abelian extensions of number fields k by means of special values of appropriately chosen analytic functions. In the simplest case, i. e. k = Q, the Kronecker-Weber theorem says that the abelian extensions of Q are completely classified by using the special values of the transcendental function z ↦ → e 2πiz at points of finite order on the circle R/Z, see [Gr]. Hence, the question of extending this theorem to any base number field k, i. e. the famous Hilbert’s 12th problem, can be formulated whether abelian extensions of k can be generated by adjoining torsion points of suitable abelian groups.