2026-04-14T08:41:57Zhttps://recercat.cat/oai/request

oai:recercat.cat:2117/1193562025-07-22T15:17:48Zcom_2072_1033col_2072_452951

Modular curves and complex multiplication Hernández Barrios, Víctor Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres Arithmetical algebraic geometry Number theory Elliptic curve Modular curve Heegner point Geometria algèbrica--Aritmètica Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry) In this project we explore the connections between elliptic curves, modular curves and complex multiplication (CM). The main theorem of CM shows that the theory of CM for elliptic curves provides an explicit construction of finite abelian extensions of a quadratic imaginary field. The proof we discuss uses many of the properties of the classical modular curve, which is introduced both as a geometrical object and as a moduli space. This theory together with the Modularity theorem is used in the construction of Heegner points, which are in the heart of the proof of Kolyvagin's theorem, a result closely related with the Birch and Swinnerton-Dyer conjecture. 2018-07 Master thesis http://creativecommons.org/licenses/by-nc-sa/3.0/es/ Open Access Universitat Politècnica de Catalunya