Elliptic Curves: Function Theory, Geometry, Arithmetic by Henry McKean (PDF)

Description

The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, whose masters were Abel, Gauss, Jacobi, and Legendre. This book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows a historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. The many exercises with hints scattered throughout the text give the reader a glimpse of further developments. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics.

User’s Reviews

Editorial Reviews: Review “…undergraduates will find that McKean and Moll’s book…offers more diverse viewpoints [than other books on elliptic curves]. Highly recommended.” Choice”…this is a wonderful book that should reward those who have the background for it with immense joy and insight.” SIAM Review Book Description An introductory 1997 account in the style of the original discoverers, treating the fundamental themes even-handedly.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The subject of elliptic curves and the corresponding elliptic integrals and elliptic functions which invert them is one of the most beautiful mathematics ever done. It brings together subjects like arithmetic, geometry and complex analysis with contributors like Abel, Gauss, Jacobi and Legendre. Prepare to enter the world of rational functions, Riemann surfaces, covering spaces, elliptic integrals and the discovery of Abel and Gauss that their inversion yields elliptic functions. Then enter the world of Jacobi’s theta functions, the modular group, the modular equation. Abel and Galois proved that the general quintic is not solved by radicals. Hermite discovered that the extra ingredient required is a new radical i.e. the eighth root of the Jacobi’s modulus. Prepare to enter the “Class field” due to Kronecker and Weber and the theorem of Mordell and André Weil that states that the rational points of elliptic curves form a module of finite rank over the whole numbers Z. Prepare to enter the world of Hyperbolic Geometry and much much more like the icosahedral group, yes definitely not an easy one but worth the effort to go through it and read it. Only for smart people!

⭐This book avoids the traps which would make this subject so inaccessible. Rather than frightening the reader with group theory and the sort of very advanced material that would fit it into a post graduate slot, the book starts with very little beyond geometry and complex number theory. The book carefully progresses to discussions on the projective line, and Riemann surfaces (never too much at once) to the inevitable subjects of the Icosohedral group, and invariant theory. It manages to do this almost without you noticing the depth of maths that is being covered – quite a feat!From here on, elliptic integrals are discussed, and the work of Jacobi, Gauss, Legendre and Abel discussed freely, with many examples and clear pictures. The text is interspersed with exercises (some of which you can do with a few moments thought, others more difficult). I enjoyed this section (and the remainder of the book) for several very interesting short accounts of subjects slightly tangential to the main material.[One of my favorites was the account of a letter with a amazingly strange but elegant identity with a continued fraction sent by Ramanujan to Hardy, and Hardy’s subsequent absolute amazement… You MUST NOT miss reading that, even if it isn’t what you picked the book up for!]Then the book goes into the area I bought the book for – modular groups, and the solution of the Quintic. This subject draws mostly on work by Hermite, and later, Klein, but is presented carefully and slowly.I was very glad to find this book. It doesn’t race through the subject at breakneck speed, which is what some books on Galois Theory or Algebraic Curves do, and has illuminated quite a few additional topics for me. I guess that now I will be able to recognize the origins of so much hard maths now (and all those entries in the tables of integrals I never understood)After all, this subject is now very important. Elliptic curves occur in many subjects – Cryptography, Information Theory, and of course, the proof of Fermats last theorem.

⭐The popular press leaves us with the impression that math isintimidating. This wasn’t always the case. In my time, the approach to how we teach math, and write books about it, went through a number of cycles, or trends; some of them now discredited;–or not!? Here is a sample: (1) I grew up with the boot-camp approach with its endless drills, (2) then came “The New-Math approach”, followed by (3) “The back-to-basics” trend. (4)Following Eric Temple Bell, it became popular for a time to mix into the teaching of math a lot of history/ or dramatic stories about the heros in the subject. Finally, more recently:(5) “The Make-it-Seem-Easy-and Fun approach” and the motivational speakers; imitating popular TV shows.—Seriously, what I like about this lovely book is that it treats mathmatics as one unified subject, and that the authors masterfully highlight a number of unexpected connections between what otherwise are thought of as isolated specialties within math: The exciting new problems are at the same time also the old and classic problems in math: The elliptic integrals of Abel and Gauss, Jacobi’s theta functions, modular functions, quadratic fields, elliptic curves, and Mordell-Weil. It is all beautifully presented. The book is selfcontained, and it is a pleasure to read. The clear and concise presentation is what makes the subject seem easy, or more importantly interesting and useful. I hope it will be a model for other math books to follow.

⭐I got this book as a gift from a long time friend. He had trouble with reading it. It is only for that reason I give it only 4 stars. These authors make others that I have read on this range of subjects look bad: Fields Medalists included! A lot of it is that they just bother to give you the real mathematics with examples. I think the initial miss definition of the Riemann surface gives a false impression, because the explanations of ramified covers and toral elliptic lattices is just wonderful. Reading this book makes Dr. Singerman’s papers look so much better! I was disappointed in the treatment of triangle groups, but the treatment of modular functions and gamma1 and gamma2 makes up for that. It is a masterful work… the best I have seen by a modern author. It reminds me of books by Ulam or Russell. Sawyer’s little book is not as good!

⭐It’s a good book but a HORRIBLE Print of Demand printing!They should tell you about this…

⭐楕円関数論と数論が渾然一体となって「虚数乗法論」に昇華する様子が活き活きと描かれ、この理論の面白さを堪能できる格好の入門書である。一方、全くの初学者向けの本ではないので、楕円関数論と代数体の基礎知識なしに、多くの演習問題をこなして最後まで読み切る事は難しいと思う。この本を少ない予備知識で読んでみたいという方は、事前に『数論への出発』（日本評論社）の第３章と第５章を一読されると良いと思う。本書は全７章からなり、１章は予備知識のまとめ、７章は楕円曲線の有理点に関するモーデル-ヴェイユの定理の解説であり、主題は２章から６章までの200頁ほどに濃密に書かれている。２章と３章は楕円関数論のコンパクトで美しい要約である。p関数とsn関数との相互関係式やp関数をヤコビのテータ関数で表示する公式などは、他書で殆ど触れられていないユニークな話題である。また、３章の終わりに「ラマヌジャンの連分数」の特殊値の導出に関する素晴らしい解説がある。４章はモジュラー群とモジュラー関数の解説である。この理論の背景に楕円積分の周期等分理論があって、（ヤコビの）モジュラー方程式とは楕円積分（周期）の変換前と変換後のモジュラスの間の関係式であり、これを利用して直接計算が難しい楕円モジュラー関数の特殊値を求めようというのである。５章はレベル５のモジュラー方程式の次数が低減できる事に着目した、エルミートによる5次方程式の解法の解説である。6章「虚2次体」は本書のハイライトであり、この数体の最大不分岐アーベル拡大体（絶対類体）が、楕円モジュラー関数の特殊値としての「イデアル類の類不変量」を用いて構成できる事が示されている。現代の達人による解説を受けながら、19世紀数学の最も華麗な王朝絵巻と現代に息づく壮麗な理論の原風景を見るような気がする素晴らしい本である。

⭐

Keywords

Free Download Elliptic Curves: Function Theory, Geometry, Arithmetic in PDF format
Elliptic Curves: Function Theory, Geometry, Arithmetic PDF Free Download
Download Elliptic Curves: Function Theory, Geometry, Arithmetic 1999 PDF Free
Elliptic Curves: Function Theory, Geometry, Arithmetic 1999 PDF Free Download
Download Elliptic Curves: Function Theory, Geometry, Arithmetic PDF
Free Download Ebook Elliptic Curves: Function Theory, Geometry, Arithmetic