J-curve

The authors do a fantastic job of introducing elliptic curves for individuals and students interested in this area. Because of the importance of elliptic curves to cryptography, in integrable models in statistical mechanics, in superstring theory in physics, in mirror symmetry in algebraic geometry, in mechanics in the solution of the spinning top, and even in financial engineering, this book will be useful in building intuition about these interesting objects. Be careful in reading this book though...the theory of elliptic curves is beautiful and addicting, and you will want no doubt to read more about them after finishing it. There are two other books by Silverman that will alleviate the monkey on your back for more knowledge about elliptic curves. Happy reading......

This wonderful book is an excellent introduction to elliptic curves over the rational numbers. It is self-contained and easily accessible, but still takes the reader quite far, thus giving an undergraduate reader some exciting glimpes of deeper mathematics. This book is ideally suited as a text book for an undergraduate course (I have myself enjoyed it as a course), but is written in a lively style that also makes it fun to read on one's own. It covers such topics as the Nagell-Lutz Theorem, Mordell's Theorem over rational numbers, elliptic curves over finite fields and reduction modulo p, Thue's Theorem and diophantine approximation, and even an introduction to complex multiplication. An appendix provides the reader with a basic background on projective geometry. This book is a must for any student wanting to see beyond the ordinary coursework, and at the same time provides a natural stepping stone to a more advanced treatment of the subject, such as "The Arithmetic of Elliptic Curves", also by Silverman, which has become pretty much the standard text on the subject.

This book is a continuation of the authors earlier book on elliptic curves, which was also an excellent book, and treats the more specialized topics in elliptic curves. I cannot think of a branch of physics or engineering that has not made use of some facet of the theory of elliptic curves, and they have myriads of applications in other fields also, such as cryptography and financial engineering. The book is very organized, straightforward to read, the author summarizes well his intentions at the beginning of each chapter, and recommends several references for topics left out of the main discussion. Space does not allow a detailed chapter by chapter review, so I will confine my review to the first two chapters, which were of main interest to me. In summary, Chapter 1 discusses how to study elliptic curves by taking a collection of them, each member being isomorphic, and studying the properties of modular functions and differential forms on this collection, now thought of as an algebraic curve, called the moduli space. The famous linear operators, called the Hecke operators, act on the the space of modular forms, and they and their eigenfunctions satisfy the same set of relations. One then attaches the well-known L-series to the modular forms that has very interesting algebraic and analytic properties. In more detail, the author does the following in the chapter. The set of lattices in the complex plane modulo non-zero multiplication L/C* is considered, along with the set of elliptic curves over the complex plane modulo complex isomorphism. These collections are proven to be bijective by showing that L/C* is isomorphic to C by first putting a complex structure on it. This leads to a surjective map from the upper-half plane H to L/C*. Proving this to be injective leads to a bijection from SL2(Z)\H to L/C*. Since the matrix -1 acts trivially on H, one can quotient out +1 and -1 and obtain the modular group. The quotient space modular group\H is a 2-sphere minus a point, but can be made into a Riemann surface by extending the upper half-plane (called H*). The modular curve X(1) = modular group\H* results and is compact and Hausdorff. A complex structure is put on it, making it into a a Riemann surface of genus 0. Meromorphic functions on X(1) are rational functions of the j function, but more interesting functions are defined on X(1), namely the modular functions, such as the Eisenstein series. These considerations lead to a proof of the uniformization theorem for elliptic curves over C. For a given elliptic curve E, a study of the set of all isogenies to E of degree n is the same as that of studying degree n maps from E to other elliptic curves, which is called the dual isogeny, and leads to the Hecke operator. The Hecke operator and the homothety operator both map the divisor group of the lattice to itself, and generate a commutative algebra, called the Hecke algebra. Hecke operators can act on modular forms of weight 2k, and modular forms exist which are simultaneous eigenfunctions for the Hecke operator of weight 2k. It can be proven, but the author does not do so, that the normalized eigenfunctions form a basis for the space of cusp forms of weight 2k. The Fourier coefficients of the eigenfunction have an Euler product decomposition of a Dirichlet series attached to f, called the L-series. In the next chapter, the author considers elliptic curves that have extra endomorphisms, called complex multiplication. The collection of endomorphisms is usually taken to be the real numbers R, or R(K), which is the ring of integers of R tensored with the rational numbers. And, just as in chapter 1, he studies collections of elliptic curves, but here ones with the same endomorphism ring., called ELL(R) in the book. Asking the question of how to construct an elliptic curve with complex multiplication by a particular R(K) leads him to studying the ideal class group of R(K), and this group is shown to act transitively on ELL(R(K)). The author also shows that every elliptic curve with complex multiplication is defined over an algebraic extension of Q. Several interesting examples of ellipti curves with complex multiplication are given. After a brief review of class field theory, the author proves that K(j(E)) is the Hilbert class field H and shows how the Galois group of H/K acts on j(E). The torsion points of E are then used to generate abelian extensions of K, using the Weber function for E/H, thus generalizing the usual cyclotomic extensions of number theory. Very interesting examples are given of these constructions and it is also shown that j(E) is an algebraic integer. Then after a brief review of cyclotomic class field theory, the author proves what he calls the main theorem of complex multiplication, which says that an automorphism of the torsion subgroup is essentially analytic multiplication by an idele of K. This theorem allows one to define a Grossencharacter associated to an elliptic curve with complex multiplication. For such a curve one can then define an L-series and show that it can be expressed as a Hecke L-series with Grossencharacter.

This book presented information regarding plane curves in a clear, concise manner. The illustrations add to the book's quality. This is a good reference for people who work with plane curves.

The book "Curves and Singularities" is an excellent introduction to the the use of calculus in studying curves and surfaces. I enjoyed the sections on osculating cirles, degree of contace, envelopes, and jets. There are many exercies and examples to guide the reader.

I saw a copy once and read the intro which alluded to removing the mystery from elliptic curve cryptography, disdaining the popular 'myth' that it's "very complicated."

Pastor J Konrad is a gift from God, every book he has ever writen has helped me to grow and become stronger. This book is no different. if you can master how to change yourself into the likeness of Christ you can master anything. I strongly encourage you to read this book, it will either help you grow or make you mad, LOL and that's the truth

This book is full of empty long winded paragraphs of hardly believable self-affirmations. It's insulting to think that this writer is helping women of size. Blow your boring hot air somewhere else. What a waste of money.

As you can see by my rating, I liked this inspirational book about self-acceptance. It focuses on plus-sized women, but most of the advice could be utilised by people who could benefit from improved self-esteem in other areas. The authors have included self-esteem-building exercises, which complement the message of the text. I particularly liked the stories from successful, well-adjusted women who wrote about their own struggles with acceptance of themselves, and by their families and the wider community - strong, uncompromising, successful women. I was disappointed, then, to find that the illustrations (photographs and drawings), almost without exception portrayed women who are average-sized or smaller. I found this particulalry inappropriate in the section of dressing to reflect your style and best features - the women drawn would have looked attractive in sackcloth! Other than this quiblle, I found this book interesting and worthwhile.

This book was written with a plus-sized audience in mind. Nevertheless, I honestly feel that the information contained within will help slightly overweight (or even normal-weight) women just as much as plus-sized ones; either way, in today's society, the reality or the specter of "excess" weight causes us emotional pain.

Since having two children, I've struggled with a weight gain of 25 pounds. Medically, at a size 12, I'm considered "overfat," between normal and obese. I've been beating myself up over my weight for a long time. I've been on many diets (never any extreme ones) and I'd have some success until I got frustrated with the "slow" results and then would find old habits, along with the weight, creeping back. In all honesty, ultimately I would still like to lose a bit of weight.

But here's the interesting thing, at least for me. In doing the exercises in the book, I slowly began to convince myself emotionally that my personal value was a thing quite apart from my weight, something I always knew intellectually, but still didn't completely believe in some corner of my mind. And a growing appreciation for the unique person that I am got me doing things like searching out high-end consignment shops to find beautiful, well-made and flattering clothes, exercising...not with the goal of losing weight, but as a way of taking care of myself..., and eating moderately, but for pleasure (only delicious food need apply for consumption). I had the self-respect to reject any possible diet/activity changes unless I could answer "yes" to the question "Am I willing to do this for the rest of my life if I never lose a pound?"

And without ever feeling like I've been "trying" to lose weight, I found my clothes getting a bit looser, got curious and discovered I've lost seven pounds, over a period of about two months. The wonderful thing is that because I've simply been focusing on taking care of myself, on a number of different levels, the result of weight loss isn't really the point. It's just a nice little bonus.

The other book I've read during this time to help me understand where the creed of thinness came from in the first place was Never Too Thin: Why Women Are at War with their Bodies by Roberta Pollack Seid, Ph.D. There were so many passages in that book that resonated with me, particularly in the chapters that dealt with the decades of my lifespan. It helped me to put the issue of weight into a more constructive perspective.

I particularly appreciated Learning Curves for encouraging women to get to know themselves, to treasure themselves and finally to take what they've applied and live, whether through quiet example or through active outreaching, as a role model for other women and young girls who haven't yet made or are just beginning their journey.

The book provides a well done empirical analysis of several different countries' data on wages and unemployment. Throughout the book, Blanchflower and Oswald confront several different theoretical problems with their picture of the relationship between unemployment and wages. Yet, some definite problems do exist in their empirical and theoretical analysis... most notably their lack of a discussion on the possibility of selection bias.

I liked the book because of their discussion and formalization of a negative relationship between unemployment and wages. Their review of the literature, possible theoretical explanations, and representation of the results are all excellent though at times rather difficult to read. The large amounts of charts and tables provide support to their idea but are at times exhausting to go through. While the book does provide several different theoretical explanations to the relationship between unemployment and wages, it does definitely lack a decisive discussion on what the wage curve actually represents.

This is a fairly complete treatment of elliptic curve cryptography. It suffers from a very uneven treatment. The chapters on implementation are well written and easy to read. The material on the logarithm problem, however, is much too advanced and will only be accessible to research mathematicians. A big omission in the book are protocols such as signatures and encryption.

This book gives a good summary of the current algorithms and methodologies employed in elliptic curve cryptography. The book is short (less than 200 pages), so most of the mathematical proofs of the main results are omitted. The authors instead concentrate on the mathematics needed to implement elliptic curve cryptography. The book is written for the reader with some experience in cryptography and one who has some background in the theory of elliptic curves. A reader coming to the field for the first time might find the reading difficult. The authors do give a brief summary in Chapter 1 on the idea of doing cryptography based on group theory. They then move on to discuss finite field arithmetic in Chapter 2. The reader is expected to know some of the basic notions of multiprecision arithmetic for integers. The authors choose to work with 2^16. Psuedocode is given for doing modular arithmetic with Montgomery arithmetic given special attention. The last section of the chapter gives a good summary of arithmetic in fields of characteristic 2. Chapter 3 discusses very compactly arithmetic in elliptic curves. This is where the reader should already have the background in the theory of elliptic curves, since the reading is very fast and formal. The authors do a good job of summarizing how modular polynomials come into play in elliptic curve cryptography and give some explicit examples of these polynomials. The most important chapter of the book is Chapter 4, where the authors give a discussion of how to implement elliptic curves efficiently in cryptosystems. This chapter is nicely written and pseudocode appears many times with lots of nice examples. This chapter serves as background for the next one on the discrete logarithm problem using elliptic curves over finite fields. The MOV attack, the anomalous attack, and the baby step/giant step methods are discussed very nicely. Random methods, such as the tame and wild kangaroo are discussed at the end of the chapter.

The next three chapters concentrate on how to actually generate elliptic curves for cryptosystems, with particular attention payed to the Schoof Algorithm. The chapter on Schoof's algorithm is more detailed than the rest of the chapters and this makes for better reading. The authors do discuss how to generate curves using complex multiplication although the discussion is somewhat hurried. The next chapter discusses how elliptic curves have been applied to other areas in cryptography, such as factoring, etc. A good discussion of the ECPP algorithm on proving primality ends the chapter. The authors end the chapter with a discussion of hyperelliptic cryptography. Anyone familiar with the theory of elliptic curves and how they are applied to cryptography will naturually ask if hyperelliptic curves have any advantages over the elliptic case. The authors never really address this explicity but do give examples on just what is involved in implementing hyperelliptic curves in cryptography. Overall a fine addition to the literature on elliptic curves in cryptography. One would hope that the authors would write a follow-up book on hyperelliptic curves and maybe on general algebraic curves and their possible use in this area.

I think this is one of the best introductions to elliptic curve cryptosystems. This book have all the last algorithms in the field.

Readers who have not yet read this book will be surprised to learn that the main topic is not race, but how intelligence explains class structure. The authors argue that intelligence, not environment is the primary determinant of a variety of social behaviors, including class, socio-economic level, crime, educational achievement, welfare, and even parental styles. Hernstein and Murray back up these claims with some of the most persuasive data ever seen in the social sciences. The importance of a person's intelligence cannot be understated. Its is the number one determinant in shaping one's life. Hernstein and Murray do not stop there however. They go on, arguing that the bottom 15 percent in intelligence are simply not capable of taking care of themselves, falling into poverty, drugs, alchoholism, etc. American society can no longer accept such conditions for lower cognitive class. They make concrete suggestions on how to change this condition. They also make striking claims about the danger of affirmative action programs in promoting people who are not qualified to do important tasks. And finally, they deal with the issue that makes this book so controversial: The lower tested intelligence of African-Americans. At no point do they the claim the gap is only due to genetics. They suggest past environmental factors come into play. But their main point is that modern day racism cannot explain the gap, and programs designed to bridge that gap will fail, and putting underqualified individuals in important positions is not the answer. The authors really do not go into detail about why the gap exists, setting themselves up for criticism. But at least another scholar can research this topic and try and explain it. In sum, this book explains class structure in America, as well as the many of the social maladies of our time. It offers proof, and conrete solutions. It is a book of monumental importance, and cannot be denounced as racist. Those who make such claims either did not read the book, or are too biased to think objectively. As Murray notes in his new afterword, modern Sociology is riddled with taboos and self-censorship. The radical leftists who dominate the field do this country a great disservice by being so biased and non-objective. They also refuse to look at biology, relying only on environmental explanations, despite pyschology's growing reliance on genetic determinants of human behavior. The general public can only hope that the field right itself. Until it does, there will no solution to our most pressing social problems.

American Enterprise Institute academic Michael Ledeen was right to call "The Bell Curve" "the most moderate book in recent years to spark such an accusatorial controversy." While it's true that "The Bell Curve" draws many surprising and concerning conclusions, and some conclusions that some people find alarming, the issues are legitimate and merit consideration.

The basic premises and theses of "The Bell Curve" are these: that intelligence, IQ, or (perhaps less inflammatorily) cognitive ability is a real, measurable, quantifiable characteristic of a human being; that different people tend to be assigned very different roles by society depending on their level of cognitive ability; that people of different cognitive abilities behave differently in some important ways; that cognitive ability is substantially heritable; and that different groups tend to have differing levels of cognitive ability. The authors support these theses using the (remarkably rich) body of literature on the subject. Their procedures are documented with great care and a tremendous variety of sources is cited.

The book can be read at a number of levels. At its shortest, the book amounts to only some thirty pages in length. Each chapter begins with a summary that briefly outlines the conclusions that will be reached. The main text of the book is about 550 pages. The content consists chiefly of validation and explanation of the authors' claims, as well as some psychometric history, all of which is both fascinating and persuasive. In addition to the primary text, the book is replete with sidenotes, endnotes, and appendices, to say nothing of the hundreds of external sources to which we are referred in the bibliography. The authors' style is simultaneously informative, accessible, frank, cautious, and persuasive.

Of especial interest to the skeptical (including me) is the afterword, in which one of the authors responds to recent critical commentary of "The Bell Curve."

Whatever your position on psychometrics and whatever critical commentary you may have read on "The Bell Curve," this book is an indispensable tool that will allow you to survey the evidence for yourself. Like many who read the primary source rather than relying on biased commentaries, you may find that the logic, not to mention the statistics, of the authors is inescapable. Regardless of your personal beliefs, the book brings up issues that will be fundamental to the future of the human race. It is at least worth your examination.

Nobody who believes that that genetics play a role in intelligence denies that enviorment does play some role also, even Hitler understood that the enviroment was important. Yet to even mention to "product of our enviroment" people that genetics plays just as big a role is regarded as blasphemy. Why is that?
This book does an excellent job at showing that genetics plays just as big a role if not bigger a role in determining intelligence then the enviorment does. Read the book and see for yourself which side has a better case.