*(English)*Zbl 0804.11039

[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 ${{\Gamma}}_{0}\left(4\right)$ 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.

##### MSC:

11G05 | Elliptic curves over global fields |

14H52 | Elliptic curves |

11Fxx | Discontinuous groups and automorphic forms |

11-02 | Research monographs (number theory) |

14-02 | Research monographs (algebraic geometry) |