Postgraduate Thesis uoadl:1321270 1047 Read counter

Τομέας Άλγεβρας Γεωμετρίας

Library of the School of Science

Library of the School of Science

2012-09-05

2012

Χατζάκος Δημήτριος

Αριστείδης Κοντογεώργης, Αναπληρωτής Καθηγητής

Modular forms και Ελλειπτικές καμπύλες

Greek

The purpose of this master thesis is to study Modular forms and Elliptic

curves. Both modular forms and elliptic curves are important areas of modern

number theory. The aim of this work is to study the deeper relationship between

these two number theoretic structures, as expressed by the Modularity theorem.

The structure of work is as follows: in the first chapter develops the basic

theory of algebraic tools needed for the study of elliptic curves, while the

second chapter studies the classical theory of geometry and arithmetic of

elliptic curves. In particular, we consider the problem of determining the

structure of E(K) for a given body K.

In the third chapter we enter the analytic part of theory. Here, the central

point of study are the upper complex half-plane and modular curves. In the

fourth chapter we define modular functions and modular forms and examine the

structure of these spaces, as well as the normal operators acting on them.

Furthermore, we develop the theory of moduli interpretation.

Finally, in the last chapter, we describe the Modularity theorem by a few

different versions.

curves. Both modular forms and elliptic curves are important areas of modern

number theory. The aim of this work is to study the deeper relationship between

these two number theoretic structures, as expressed by the Modularity theorem.

The structure of work is as follows: in the first chapter develops the basic

theory of algebraic tools needed for the study of elliptic curves, while the

second chapter studies the classical theory of geometry and arithmetic of

elliptic curves. In particular, we consider the problem of determining the

structure of E(K) for a given body K.

In the third chapter we enter the analytic part of theory. Here, the central

point of study are the upper complex half-plane and modular curves. In the

fourth chapter we define modular functions and modular forms and examine the

structure of these spaces, as well as the normal operators acting on them.

Furthermore, we develop the theory of moduli interpretation.

Finally, in the last chapter, we describe the Modularity theorem by a few

different versions.

Modular Forms, Elliptic Curves, Modularity Theorem

No

0

Yes

42

XI, 209