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Riemann surfaces and dessin d'enfants
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In this document, I have recovered what I studied during the years 2016 and 2017 with Professor Carlos A. A. Florentino. The first chapter covers the basic notions of the theory of Riemann Surfaces, some important results such as the Euler-Poincaré characteristic, or the Hurwitz formula, the definition of bundles and sheaves and a proof of the Riemann-Roch theorem. This theory is used in the second chapter to proof the main theorem of this thesis, Belyi’s theorem:
”A compact Riemann surface S is defined over a number field if and only if there exists a Belyi map on S”
The proof requires us to introduce some results of ”valuations” and ”specializations”. In the third chapter, we give the first definitions and results of dessins d’enfants, and how they are related to the Riemann surfaces (and so, to the algebraic curves). We end the thesis giving some important examples and results of a more particular nature, related to the theory of dessins d’enfants, such as the Shabat polynomials.
Descrição
Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, em 2018
Palavras-chave
Riemann surfaces Dessind’enfants Cohomology Riemann Teses de mestrado - 2018