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- Type of Document: Ph.D. Dissertation
- Language: Farsi
- Document No: 49963 (02)
- University: Sharif University of Technology
- Department: Mathematical Sciences
- Advisor(s): Fanai, Hamid Reza; Shahshahani, Mehrdad
- Abstract:
- The main purpose of this thesis is to study computational questions related to compact Riemann surfaces (algebraic curves) with main emphasis on the theory of Grothendieck’s dessins d’enfants; where recovering the explicit formula of a Belyi map from the corresponding dessin on a topological surface is important. Some applications such as investigating the modular j-function and also certain classes of modular curves are included in this thesis along with a summary of attempts toward generalizing the Belyi theorem to complex dimension two.First two chapters contain necessary prerequisites on compact Riemann surfaces and a short introduction to the theory of dessins d’enfants and Belyi maps.In lower genera, many calculations can be carried out explicitly and fairly simple descriptions of the corresponding moduli spaces may be obtained. In the third chapter,after reviewing the parametrization of the moduli space in genus one or two with j-invariant and Igusa’s invariants respectively, we will derive algebraic formulas for certain isogenies between elliptic curves or morphisms from a genus two curve onto an elliptic curve which will be employed to derive several special values of the function j, few identities for this function and to study periods of genus two Riemann surfaces with large groups of automorphisms. In genus three, there is a discussion on Hurwitz spaces. We will use the classical invariant theory of finite groups throughout this chapter to describe these moduli and Hurwitz spaces.Going back to dessins in the fourth chapter, we shall explain how a class of modular curves are naturally equipped with Belyi functions. After exhibiting few examples of this kind, we will focus on one of the theorems proven in this thesis which classifies Riemann surfaces equipped with certain pairs of Belyi functions as modular curves.The main example is the modular curve X0(N) that, as instances of this theorem, has been studied thoroughly with drawing dessins and writing the corresponding modular equations for values N = 2; 3; 4. The last chapter is a brief introduction to works trying to find an analog of Belyi’s theorem in complex dimension two along with topological and analytic difficulties that emerge when one tries to formulate a suitable generalization
- Keywords:
- Belyi Map ; Dessin Denfant Theory ; Moduli Space ; Riemann Surface ; Ramified Covering ; J-Invarient ; Modular Curve ; Igusa's Invarient
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