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Introduction to Elliptic Curves and Modular Forms

Neal Koblitz
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 97
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The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This book takes the "complex variables" view of elliptic curves. It uses a particular number-theoretic problem to drive the discussion: the problem of characterizing congruent numbers. A congruent number (no relation to congruences modulo a number) is defined as a number that is the area of a right triangle with all sides rational. Thus 6 is a congruent number because it is the area of the familiar 3-4-5 right triangle. There is an equivalent formulation in terms of whether the elliptic curve y2 = x3 - n2x has rational solutions. This enables us to bring the powerful machinery of elliptic curves to bear on the problem, and the book develops that machinery and culminates with a proof of Tunnell's (almost complete) characterization of congruent numbers.

I like the method of using a single difficult program to organize a book. I think it is not completely successful here, because the original problem drops out of view in the middle of the book, with many new concepts being introduced that are not clearly driven by it. The book travels though L and zeta funtions, elliptic functions, and modular functions and forms.

Silverman and Tate's Rational Points on Elliptic Curves is a very different approach to elliptic curves, through abstract algebra and geometry. There is surprisingly little overlap between the two books, considering that they are introductions to the same subject. Koblitz is much faster-paced, and contains a lot of intricate arguments. It covers a much larger amount of material and requires more mathematical maturity (it is correctly placed in Springer's Graduate Texts series, while Silverman and Tate is in the Undergraduate Texts series). I like both books, but I think Silverman and Tate is a better introduction.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.



Preface to the First Edition

Preface to the Second Edition

CHAPTER I From Congruent Numbers to Elliptic Curves
1. Congruent numbers
2. A certain cubic equation
3. Elliptic curves
4. Doubly periodic functions
5. The field of elliptic functions
6. Elliptic curves in Weierstrass form
7. The addition law
8. Points of finite order
9. Points over finite fields, and the congruent number problem

CHAPTER II The Hasse-Weil L-Function of an Elliptic Curve
1. The congruence zeta-function
2. The zeta-function of En
3. Varying the prime p
4. The prototype: the Riemann zeta-function
5. The Hasse-Weil L-function and its functional equation
6. The critical value

CHAPTER III Modular forms
1. SL2(Z) and its congruence subgroups
2. Modular forms for SL2(Z)
3. Modular forms for congruence subgroups
4. Transformation formula for the theta-function
5. The modular interpretation, and Hecke operators

CHAPTER IV Modular Forms of Half Integer Weight
1. Definitions and examples
2. Eisenstein series of half integer weight for ̃Γ 0(4)
3. Hecke operators on forms of half integer weight
4. The theorems of Shimura, Waldspurger, Tunnell, and the congruent number problem

Answers, Hints, and References for Selected Exercises