##
**Introduction to elliptic curves and modular forms.
2. ed.**
*(English)*
Zbl 0804.11039

Graduate Texts in Mathematics. 97. New York: Springer-Verlag. x, 248 p. (1993).

[For a review of the first edition (1984) see Zbl 0553.10019).]

The book covers basic properties of elliptic curves and modular forms. As a motivating example to study these subjects the ancient “congruent number problem” is taken (recall that a rational number \(n\) is called “congruent” if it is the area of some right triangle with rational sides).

Concerning elliptic curves the author discusses, in particular, the addition law, points of finite order and the Hasse-Weil \(L\)-function, always with special emphasis of those curves attached to congruent numbers.

In Chap. III basic concepts on modular forms of integral weight are studied. In Chap. IV the author discusses Shimura’s theory of modular forms of half-integral weight. An explicit computation of the Fourier expansion of the Eisenstein series on \(\Gamma_ 0 (4)\) is given, Hecke operators and the Shimura lift are investigated and Waldspurger’s theorem on the critical values at the center of the twists of the \(L\)-series of a Hecke eigenform of integral weight in some special cases is stated.

The book concludes with a characterization of a squarefree positive congruent number through the number of representations of \(n\) by certain ternary quadratic forms, a result of Tunnell (1983). The proof uses most of the material covered in the previous sections of the book.

The book includes lots of exercises, with some answers and hints for their solutions, and is very pleasant to read. As the author remarks, since the appearance of the first edition there had been some major progress in the solution of outstanding questions in the theory of elliptic curves (e.g. in the direction of the conjecture of Birch and Swinnerton-Dyer), and the second edition wants to update the bibliography and the current state of knowledge of the arithmetic of elliptic curves.

The book covers basic properties of elliptic curves and modular forms. As a motivating example to study these subjects the ancient “congruent number problem” is taken (recall that a rational number \(n\) is called “congruent” if it is the area of some right triangle with rational sides).

Concerning elliptic curves the author discusses, in particular, the addition law, points of finite order and the Hasse-Weil \(L\)-function, always with special emphasis of those curves attached to congruent numbers.

In Chap. III basic concepts on modular forms of integral weight are studied. In Chap. IV the author discusses Shimura’s theory of modular forms of half-integral weight. An explicit computation of the Fourier expansion of the Eisenstein series on \(\Gamma_ 0 (4)\) is given, Hecke operators and the Shimura lift are investigated and Waldspurger’s theorem on the critical values at the center of the twists of the \(L\)-series of a Hecke eigenform of integral weight in some special cases is stated.

The book concludes with a characterization of a squarefree positive congruent number through the number of representations of \(n\) by certain ternary quadratic forms, a result of Tunnell (1983). The proof uses most of the material covered in the previous sections of the book.

The book includes lots of exercises, with some answers and hints for their solutions, and is very pleasant to read. As the author remarks, since the appearance of the first edition there had been some major progress in the solution of outstanding questions in the theory of elliptic curves (e.g. in the direction of the conjecture of Birch and Swinnerton-Dyer), and the second edition wants to update the bibliography and the current state of knowledge of the arithmetic of elliptic curves.

Reviewer: W.Kohnen (Bonn)

### MSC:

11G05 | Elliptic curves over global fields |

14H52 | Elliptic curves |

11Fxx | Discontinuous groups and automorphic forms |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

### Keywords:

congruent number problem; Hasse-Weil \(L\)-function; elliptic curves; modular forms; integral weight; half-integral weight; Fourier expansion; Hecke operators; Shimura lift; Waldspurger’s theorem### Citations:

Zbl 0553.10019
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\textit{N. Koblitz}, Introduction to elliptic curves and modular forms. 2. ed. New York: Springer-Verlag (1993; Zbl 0804.11039)

### Digital Library of Mathematical Functions:

§20.10(i) Mellin Transforms with respect to the Lattice Parameter ‣ §20.10 Integrals ‣ Properties ‣ Chapter 20 Theta Functions§20.12(i) Number Theory ‣ §20.12 Mathematical Applications ‣ Applications ‣ Chapter 20 Theta Functions

§20.9(iii) Riemann Zeta Function ‣ §20.9 Relations to Other Functions ‣ Properties ‣ Chapter 20 Theta Functions

§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem ‣ §22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic Functions

Other Notations ‣ §23.1 Special Notation ‣ Notation ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.20(ii) Elliptic Curves ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.20(v) Modular Functions and Number Theory ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

Chapter 23 Weierstrass Elliptic and Modular Functions

### Online Encyclopedia of Integer Sequences:

a(n) = |E(GF(p))| - (p+1) where E(GF(p)) is the group of rational points on the elliptic curve E: y^2 + y = x^3 - x^2 mod p and the prime p is p(n) or p(n+1) according as n < 5 or n >= 5.a(n) = |E(GF(p))| = number of rational points on the elliptic curve E: y^2 + y = x^3 - x^2 mod p where the prime p is p(n) or p(n+1) according as n < 5 or n >= 5.

a(n) = |E(GF(p))/H| where E(GF(p)) is the group of rational points on the elliptic curve E: y^2 + y = x^3 - x^2 mod p, the prime p is prime(n) or prime(n+1) according as n < 5 or n >= 5 and H = {oo, (0,0), (0,-1), (1,0), (1,-1)}.