Congruences between modular forms
This master s thesis is intended to give a presentation of the theory of congruences between the Fourier coecients of modular forms. In order to do that we introduce the reader to the basic theory of modular forms from the beginning and we study the structure of their Fourier coecients in diV...
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| Tipo de recurso: | tesis de maestría |
| Fecha de publicación: | 2018 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/119359 |
| Acceso en línea: | https://hdl.handle.net/2117/119359 |
| Access Level: | acceso abierto |
| Palabra clave: | Automorphic forms Discontinuous groups Number Theory Modular Forms Congruences between modular forms Formes automòrfiques Grups discontinus Classificació AMS::11 Number theory::11F Discontinuous groups and automorphic forms Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres |
| Sumario: | This master s thesis is intended to give a presentation of the theory of congruences between the Fourier coecients of modular forms. In order to do that we introduce the reader to the basic theory of modular forms from the beginning and we study the structure of their Fourier coecients in di↵erent ways using Hecke operators. Then we start the theory of congruences finding some of them by classical methods of Number Theory. After that, we introduce the advances made by Swinnerton-Dyer in the study of congruences using l-adic representations and the generalisation by Ken Ono. Finally, we explain the papers by Hida and Ribet in two chapters giving some conditions for the existence of congruences using the associated L-functions and decomposing the space of modular forms. |
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