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**Elliptic curves. With an appendix by Ruth Lawrence.**
*(English)*
Zbl 0605.14032

Graduate Texts in Mathematics, 111. New York etc.: Springer-Verlag. XV, 350 p. DM 110.00 (1987).

The book under review gives an introduction to the theory of elliptic curves over number-fields, starting from the very elementary and ending up with an account of the newest development. To describe its contents in more detail we might divide it somehow into two parts:

The first part deals with cubic equations. More generally complete proofs are given, and the material can be understood (or at least should be) by a good undergraduate. The material covered includes plane algebraic curves, the chord-tangent law for composition on cubic curves, classification of elliptic curves up to isomorphism, some explicit families of elliptic curves, reduction mod \(p\), and Mordell’s theorem about finite generation of the group of rational points.

The second part might be called “highlights of the last 50 years of elliptic curve theory”. Here the style is more narrative, and instead of proofs we often get references to the literature. That this has to be so can be seen from the contents: Galois-cohomology, elliptic and hypergeometric functions, theta functions, modular functions, endomorphisms of elliptic curves, elliptic curves over finite and local fields, elliptic curves over global fields and \(\ell\)-adic representations, \(L\)-functions, Birch and Swinnerton-Dyer conjecture.

All in all the book is well written, and can serve as basis for a student seminar on the subject. It is however clear that reading the book is not sufficient to become an expert in the arithmetic theory of elliptic curves. This still requires skills in many more fields, from schemes to class field theory.

The book contains many exercises (mostly in the first part), and there is an appendix (by Ruth Lawrence) where they are solved.

The first part deals with cubic equations. More generally complete proofs are given, and the material can be understood (or at least should be) by a good undergraduate. The material covered includes plane algebraic curves, the chord-tangent law for composition on cubic curves, classification of elliptic curves up to isomorphism, some explicit families of elliptic curves, reduction mod \(p\), and Mordell’s theorem about finite generation of the group of rational points.

The second part might be called “highlights of the last 50 years of elliptic curve theory”. Here the style is more narrative, and instead of proofs we often get references to the literature. That this has to be so can be seen from the contents: Galois-cohomology, elliptic and hypergeometric functions, theta functions, modular functions, endomorphisms of elliptic curves, elliptic curves over finite and local fields, elliptic curves over global fields and \(\ell\)-adic representations, \(L\)-functions, Birch and Swinnerton-Dyer conjecture.

All in all the book is well written, and can serve as basis for a student seminar on the subject. It is however clear that reading the book is not sufficient to become an expert in the arithmetic theory of elliptic curves. This still requires skills in many more fields, from schemes to class field theory.

The book contains many exercises (mostly in the first part), and there is an appendix (by Ruth Lawrence) where they are solved.

Reviewer: G. Faltings