*(English)*Zbl 0911.14015

The monograph is a continuation of the same author’s book: *J. Silverman*, “The arithmetic of elliptic curves” (1986; Zbl 0585.14026). However it can be used independently from this book, on the basis of some familiarity with the basic theory of elliptic curves.

The first chapter treats elliptic and modular functions on the basis of classical complex analysis. Jacobi’s product formula for ${\Delta}\left(\tau \right)$, the action of Hecke-operators on modular forms and the $L$-series attached to modular forms are included. The second chapter is “Complex multiplication”, including a brief review of class field theory and ends with a detailed study of the $L$-series attached to a $C\text{M}$ elliptic curve. The third chapter is about elliptic surfaces. It includes a brief introduction to the geometry of algebraic surfaces (intersection theory and minimal models) (not with complete proofs). It culminates in the proof of the Mordell-Weil theorem for function fields, construction of the canonical height, and specialization theorems for the canonical height. The fourth chapter is a treatment of the Néron-model. In a sense this is a continuation of the previous chapter, based on the language of schemes, since it treats the case of arithmetic surfaces. It gives a definition of the Néron-model, and studies the fibres (Kodaira-Néron classification). It also includes an introduction to intersection theory on arithmetic surfaces. Also, Ogg’s formula on the relation between the conductor, the minimal discriminant, and the special fibre of the minimal model (for a local field) is verified. The last two chapters treat topics for curves over local fields, namely the $p$-adic uniformization, the Tate-curve and Serre’s proof for the integrality of the $j$-invariant of a CM-curve (chapter V), and local height functions (chapter VI).

The book is well written. It covers a large part of a fascinating and advanced development in mathematics, combining (complex) analysis, geometry and number theory. It keeps the needed advanced technology on a level absolutely necessary for the treatment of the subject but well enough explained for the actual purposes. Thus the book could be recommended for a wide audience to become acquainted with this fascinating topic.

##### MSC:

14H52 | Elliptic curves |

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

11-02 | Research monographs (number theory) |

11Gxx | Arithmetic algebraic geometry (Diophantine geometry) |

14-02 | Research monographs (algebraic geometry) |

11G07 | Elliptic curves over local fields |

11G05 | Elliptic curves over global fields |

14G10 | Zeta-functions and related questions |