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An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects - elliptic curves and modular forms - come together in Eichler-Shimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem.Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematics - including class field theory, arithmetic algebraic geometry, and group representations - in which the concidence of L functions relates analysis and algebra in the most fundamental ways.Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern Eichler-Shimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains Eichler-Shimura theory in such an accessible manner.

## About book:

## About file:

Security code:

- Series:
**MN-40** - Author:
**Anthony W. Knapp** - Year:
**1992** - Publisher:
**Princeton University Press** - Language:
**English** - ISBN:
**9780691085593,0691085595**

- File size:
**25 561 898** - Format:
**djvu**

Security code:

This book surveys some recent developments in the arithmetic of modular elliptic curves. It places special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modular...

This book provides a comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms. The construction of Galois representations, which play a fundamental role in Wiles' proof of the Shimur...

The study of special cases of elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centers of research in number theory. This book, addressed to beginning graduate students, introduces basic theory from a contempo...

Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition ...

This book is meant to be an introductory text, albeit at an upper graduate level. The main prerequisite for reading this book is some familiarity with the basic theory of elliptic curves as described, for example, in the first volume. Numerous exercises h...

This book is an introduction to the theory of elliptic curves, ranging from its most elementary aspects to current research. The first part, which grew out of Tate's Haverford lectures, covers the elementary arithmetic theory of elliptic curves over the ...

Elliptic curves are of central and growing importance in computational number theory, with numerous applications in such areas as cryptography, primality testing and factorisation. This book, now in its second edition, presents a thorough treatment of man...

This volume contains the expanded versions of the lectures given by the authors at the C. I. M. E. instructional conference held in Cetraro, Italy, from July 12 to 19, 1997. The papers collected here are broad surveys of the current research in the arithm...

Koblitz is in his element with this text. Much like Daniel Marcus's Number Fields, Koblitz develops a ground work to begin the study of elliptic curves. Here he builds upon the ancient problem of congruent numbers to help develop motivation for an in de...

The Third Edition of this standard reference work has grown to keep pace with our expanding knowledge of this vital subject, and will now consist of five volumes. Most chapters which appeared in the Second Edition have been revised. New chapters concern t...

The Second Edition of this popular textbook has benefited from several years of exposure to both teachers and students. Based on their own experiences as well as those of others, the authors have reorganized, added, and updated this work to meet the needs...

''Pigment-protein Complexes in Plastics - Synthesis and Assembly'' covers various aspects of the biosynthesis, assembly and function of plastid pigment-protein complexes. Its major focus is on chlorophyll-protein complexes which are addressed in relation ...