Introduction to Elliptic Curves and Modular Forms

verfasst von: Neal Koblitz

Verlag: Springer US

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

Über dieses Buch

This textbook covers the basic properties of elliptic curves and modular forms, with emphasis on certain connections with number theory. The ancient "congruent number problem" is the central motivating example for most of the book. My purpose is to make the subject accessible to those who find it hard to read more advanced or more algebraically oriented treatments. At the same time I want to introduce topics which are at the forefront of current research. Down-to-earth examples are given in the text and exercises, with the aim of making the material readable and interesting to mathematicians in fields far removed from the subject of the book. With numerous exercises (and answers) included, the textbook is also intended for graduate students who have completed the standard first-year courses in real and complex analysis and algebra. Such students would learn applications of techniques from those courses, thereby solidifying their under standing of some basic tools used throughout mathematics. Graduate stu dents wanting to work in number theory or algebraic geometry would get a motivational, example-oriented introduction. In addition, advanced under graduates could use the book for independent study projects, senior theses, and seminar work.

Inhaltsverzeichnis

Frontmatter

Chapter I. From Congruent Numbers to Elliptic Curves

Abstract

The theory of elliptic curves and modular forms is one subject where the most diverse branches of mathematics come together: complex analysis, algebraic geometry, representation theory, number theory. While our point of view will be number theoretic, we shall find ourselves using the type of techniques that one learns in basic courses in complex variables, real variables, and algebra. A well-known feature of number theory is the abundance of conjectures and theorems whose statements are accessible to high school students but whose proofs either are unknown or, in some cases, are the culmination of decades of research using some of the most powerful tools of twentieth century mathematics

Neal Koblitz

Chapter II. The Hasse—Weil L-Function of an Elliptic Curve

Abstract

At the end of the last chapter, we used reduction modulo p to find some useful information about the elliptic curves E_n: y² = x³ − n²x and the congruent number problem. We considered E_n as a curve over the prime field \({{\mathbb{F}}_{p}} \) where \( p\nmid 2n \) used the easily proved equality \( \# {{E}_{n}}({{\mathbb{F}}_{p}}) = p + 1 \) when p = 3 (mod 4); and, by making use of infinitely many such p, were able to conclude that the only rational points of finite order on E_n are the four obvious points of order two. This then reduced the congruent number problem to the determination of whether r, the rank of \( {{E}_{n}}(\mathbb{Q}) \), is zero or greater than zero

Neal Koblitz

Chapter III. Modular Forms

Abstract

Our treatment of the introductory material will be similar to that in Serre’s A Course in Arithmetic (Chapter VII), except that we shall bring in the “level” from the very beginning

Neal Koblitz

Chapter IV. Modular Forms of Half Integer Weight

Abstract

Let k be a positive odd integer, and let λ = (k − 1)/2. In this chapter we shall look at modular forms of weight k/2 = λ + 1/2, which is not an integer but rather half way between two integers. Roughly speaking, such a modular form f should satisfy f((az + b)/(cz + d)) = (cz + d)^λ+1/2f(z) for \( \left( {\begin{array}{*{20}{c}} a & b \\ c & d \\ \end{array} } \right) \) in Γ = SL₂(ℤ) or some congruence subgroup Γ′ ⊂= Γ. However, such a simple- minded functional equation leads to inconsistencies (see below), basically because of the possible choice of two branches for the square root. A subtler definition is needed in order to handle the square root properly. One must introduce a quadratic character, corresponding to some quadratic extension of ℚ. Roughly speaking, because of this required “twist” by a quadratic character, the resulting forms turn out to have interesting relationships to the arithmetic of quadratic fields (such as L-series and class numbers). Moreover, recall that the Hasse-Weil L-series for our family of elliptic curves E_n: y² = x³ − n²x in the congruent number problem involved “twists” by quadratic characters as n varies (see Chapter II). It turns out that the critical values L(E_n, 1) for this family of L-series are closely related to certain modular forms of half-integral weight

Neal Koblitz

Backmatter

Titel: Introduction to Elliptic Curves and Modular Forms
verfasst von: Neal Koblitz
Verlag: Springer US
Electronic ISBN: 978-1-4684-0255-1
Print ISBN: 978-1-4684-0257-5
DOI: https://doi.org/10.1007/978-1-4684-0255-1

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter I. From Congruent Numbers to Elliptic Curves

Chapter II. The Hasse—Weil L-Function of an Elliptic Curve

Chapter III. Modular Forms

Chapter IV. Modular Forms of Half Integer Weight

Backmatter