Author: Ahmad El-GuindyPublisher:
ISBN:
Category :
Languages : en
Pages : 78
Book Description
Weierstrass Points on Modular Curves
Author: Ahmad El-GuindyRational Points on Modular Elliptic Curves
Author: Henri DarmonPublisher: American Mathematical Soc.
ISBN: 0821828681
Category : Mathematics
Languages : en
Pages : 146
Book Description
The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 7-9, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and Swinnerton-Dyer conjecture.
Drinfeld Modular Forms Modulo a Prime Ideal and Weierstrass Points on Drinfeld Modular Curves
Author: Publisher:
ISBN:
Category :
Languages : en
Pages : 190
Book Description
We first consider the action of the Hasse derivatives on Drinfeld modular forms, which were shown by Uchino and Satoh to act as differential operators on the algebra of Drinfeld quasi-modular forms. While these operators do not preserve modularity, we show that they do preserve modularity modulo a prime ideal. We also study the behavior of the filtration under the action of the first Hasse derivative, and obtain results analogous to those obtained by Serre and Swinnerton-Dyer about Ramanujan's Theta-operator in the classical setting. We then consider a family of modular curves constructed by Drinfeld, and we study their Weierstrass points, a finite set of points of geometric interest. These curves are moduli spaces for Drinfeld modules with level structure, which are the objects which in our setting play a role analogous to that of elliptic curves. Previous work of Baker shows that for each Weierstrass point of these curves, the reduction modulo a certain prime ideal of the underlying Drinfeld module is supersingular. We study a modular form W for this congruence subgroup whose divisor is closely related to the set of Weierstrass points, an idea first presented by Rohrlich in the classical setting. To this end, we first establish a one-to-one correspondence between certain Drinfeld modular forms on the congruence subgroup and forms on the full modular group. In certain cases we can then use knowledge about the action of the Hasse derivatives to compute explicitly a form that is congruent to W modulo our prime ideal. This allows us to obtain an analogue of Rohrlich's result, which is the first important step towards obtaining a more precise relationship between the supersingular locus and Weierstrass points on our modular curves, as illustrated by Ahlgren and Ono in the classical setting.
Torsion Points on Modular Curves
Author: Matthew Howard BakerPublisher:
ISBN:
Category :
Languages : en
Pages : 190
Book Description
Introduction to Elliptic Curves and Modular Forms
Author: Neal I. KoblitzPublisher: Springer Science & Business Media
ISBN: 1461209099
Category : Mathematics
Languages : en
Pages : 262
Book Description
The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book starts out with a problem from elementary number theory and proceeds to lead its reader into the modern theory, covering such topics as the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. This new edition details the current state of knowledge of elliptic curves.
Galois Theory and Modular Forms
Author: Ki-ichiro HashimotoPublisher: Springer Science & Business Media
ISBN: 1461302498
Category : Mathematics
Languages : en
Pages : 392
Book Description
This volume is an outgrowth of the research project "The Inverse Ga lois Problem and its Application to Number Theory" which was carried out in three academic years from 1999 to 2001 with the support of the Grant-in-Aid for Scientific Research (B) (1) No. 11440013. In September, 2001, an international conference "Galois Theory and Modular Forms" was held at Tokyo Metropolitan University after some preparatory work shops and symposia in previous years. The title of this book came from that of the conference, and the authors were participants of those meet All of the articles here were critically refereed by experts. Some of ings. these articles give well prepared surveys on branches of research areas, and many articles aim to bear the latest research results accompanied with carefully written expository introductions. When we started our re~earch project, we picked up three areas to investigate under the key word "Galois groups"; namely, "generic poly nomials" to be applied to number theory, "Galois coverings of algebraic curves" to study new type of representations of absolute Galois groups, and explicitly described "Shimura varieties" to understand well the Ga lois structures of some interesting polynomials including Brumer's sextic for the alternating group of degree 5. The topics of the articles in this volume are widely spread as a result. At a first glance, some readers may think this book somewhat unfocussed.
On Weierstrass Points for Elliptic Modular Forms
Author: Edward William ChillakPublisher:
ISBN:
Category :
Languages : en
Pages : 158
Book Description
The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $q$-series
Author: Ken OnoPublisher: American Mathematical Soc.
ISBN: 0821833685
Category : Mathematics
Languages : en
Pages : 226
Book Description
Chapter 1.
Elliptic Curves, Modular Forms, and Their L-functions
Author: Álvaro Lozano-RobledoPublisher: American Mathematical Soc.
ISBN: 0821852426
Category : Mathematics
Languages : en
Pages : 217
Book Description
Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.
Algebraic Geometry and Arithmetic Curves
Author: Qing LiuPublisher: Oxford University Press
ISBN: 0191547808
Category : Mathematics
Languages : en
Pages : 593
Book Description
This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.