nach oben

2009 | Buch | 2. Auflage

Kapitel lesen Erstes Kapitel lesen

The Arithmetic of Elliptic Curves

verfasst von: Joseph H. Silverman

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Inhaltsverzeichnis

Frontmatter

Chapter I. Algebraic Varieties

Abstract

In this chapter we describe the basic objects that arise in the study of algebraic geometry. We set the following notation, which will be used throughout this book.

Joseph H. Silverman

Chapter II. Algebraic Curves

Abstract

In this chapter we present basic facts about algebraic curves, i.e., projective varieties of dimension one, that will be needed for our study of elliptic curves. Actually, since elliptic curves are curves of genus one, one of our tasks will be to define the genus of a curve. As in Chapter I, we give references for those proofs that are not included. There are many books in which the reader will find more material on the subject of algebraic curves, for example[111, Chapter IV], [133], [180], [243], [99, Chapter 2], and [302].

Joseph H. Silverman

Chapter III. The Geometry of Elliptic Curves

Abstract

Elliptic curves, our principal object of study in this book, are curves of genus one having a specified base point. Our ultimate goal, as the title of the book indicates, is to study the arithmetic properties of these curves. In other words, we will be interested in analyzing their points defined over arithmetically interesting fields, such as finite fields, local (p-adic) fields, and global (number) fields. However, before doing so we are well advised to study the properties of these curves in the simpler situation of an algebraically closed field, i.e., to study their geometry. This reflects the general principle in Diophantine geometry that in attempting to study any significant problem, it is essential to have a thorough understanding of the geometry before one can hope to make progress on the number theory. It is the purpose of this chapter to make an intensive study of the geometry of elliptic curves over arbitrary algebraically closed fields. (The particular case of elliptic curves over the complex numbers is studied in more detail in Chapter VI.)

Joseph H. Silverman

Chapter IV. The Formal Group of an Elliptic Curve

Abstract

Let E be an elliptic curve. In this chapter we study an “infinitesimal” neighborhood of E centered at the origin O.

Joseph H. Silverman

Chapter V. Elliptic Curves over Finite Fields

Abstract

In this chapter we study elliptic curves defined over a finite field \(\mathbb{F}_{q}\). The most important arithmetic quantity associated to such a curve is its number of rational points.

Joseph H. Silverman

Chapter VI. Elliptic Curves over C

Abstract

Evaluation of the integral giving arc length on a circle, namely \(\int dx/\sqrt{1 - x^{2}}\), leads to an inverse trigonometric function. The analogous problem for the arc length of an ellipse yields an integral that is not computable in terms of so-called elementary functions.

Joseph H. Silverman

Chapter VII. Elliptic Curves over Local Fields

Abstract

In this chapter we study the group of rational points on an elliptic curve defined over a field that is complete with respect to a discrete valuation. We start with some basic facts concerning Weierstrass equations and “reduction modulo π.” This enables us to break our problem into several pieces, and then, by examining each piece individually, to deduce a great deal about the group of rational points as a whole.

Joseph H. Silverman

Chapter VIII. Elliptic Curves over Global Fields

Abstract

Let K be a number field and let E∕K be an elliptic curve. Our primary goal in this chapter is to prove the following result.

Joseph H. Silverman

Chapter IX. Integral Points on Elliptic Curves

Abstract

Many elliptic curves have infinitely many rational points, although the Mordell–Weil theorem assures us that the group of rational points is finitely generated. Another natural Diophantine question is that of determining how many of the rational points on a given (affine) Weierstrass equation have integral coordinates. In this chapter we prove a theorem of Siegel that says that there are only finitely many such integral points. Siegel gave two proofs of his theorem, which we present in (IX §3) and (IX §4). Both proofs make use of techniques from the theory of Diophantine approximation, and thus do not provide an effective procedure for actually finding all of the integral points. However, Siegel’s second proof reduces the problem to that of solving the so-called unit equation, which in turn can be effectively resolved using methods from transcendence theory. We discuss effective solutions, without giving proofs, in (IX §5).

Joseph H. Silverman

Chapter X. Computing the Mordell–Weil Group

Abstract

A better title for this chapter might be “Computing the Weak Mordell–Weil Group,” since we will be concerned solely with the problem of computing generators for the group E(K)∕mE(K).

Joseph H. Silverman

Chapter XI. Algorithmic Aspects of Elliptic Curves

Abstract

The burgeoning field of computational number theory asks for practical algorithms to compute solutions to arithmetic problems. For example, the Mordell–Weil theorem (VIII.6.7) says that the group of rational points on an elliptic curve is finitely generated, and although we still lack an effective algorithm that is guaranteed to find a set of generators, there are algorithms that often work well in practice. Similarly, Siegel’s theorem (IX.3.2.1) says that an elliptic curve has only finitely many S-integral points, but it took 50 years from Siegel’s proof of finiteness to Baker’s theorem giving an effective bound for the height of the largest solution (IX §5). And Baker’s theorem is only the beginning of the story, since it leads to estimates that, although effective, are not practical without the introduction of significant additional ideas.

Joseph H. Silverman

Backmatter

Titel: The Arithmetic of Elliptic Curves
verfasst von: Joseph H. Silverman
Verlag: Springer New York
Electronic ISBN: 978-0-387-09494-6
Print ISBN: 978-0-387-09493-9
DOI: https://doi.org/10.1007/978-0-387-09494-6