##
**One semester of elliptic curves.**
*(English)*
Zbl 1083.11001

EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society Publishing House (ISBN 3-03719-015-9/pbk). viii, 130 p. (2006).

This book offers an introduction to the theory of elliptic curves based on the analytic approach. The content of the chapters is described by the keywords elliptic integrals, elliptic functions, elliptic curves, projective coordinates, group law (associativity is proved for fields of arbitrary characteristic by proving it first over the complex numbers and then arguing that associativity gives a polynomial relation which then is valid over any field), \(j\)-function, finite fields, division polynomials, torsion points over algebraically closed fields, complex multiplication, and modular forms. In contrast to the book by J. Silverman and J. Tate [Rational points on elliptic curves. New York: Springer (1992; Zbl 0752.14034)], which is aimed at a similar audience, the book under review does not discuss number-theoretical results like the theorems of Mordell-Weil or Nagell-Lutz.

The prerequisites are modest: it helps to be familiar with the basics of complex analysis, and occasional remarks involving results from topology, Galois theory, or algebraic number theory may safely be disregarded by absolute beginners. The language is a bit bumpy at some places, and the style is still very close to lecture notes: some sections (and even chapters) do not contain a single lemma, proposition or theorem. The text contains more than 100 exercises in which the reader is usually asked to complete certain parts of the proofs.

Overall, this is a very nice introduction to elliptic curves; although the approach is analytic, it is useful also for computer scientists interested in cryptographic applications because techniques needed for point counting (division polynomials) or for constructing curves with certain properties (complex multiplication) are discussed here.

The prerequisites are modest: it helps to be familiar with the basics of complex analysis, and occasional remarks involving results from topology, Galois theory, or algebraic number theory may safely be disregarded by absolute beginners. The language is a bit bumpy at some places, and the style is still very close to lecture notes: some sections (and even chapters) do not contain a single lemma, proposition or theorem. The text contains more than 100 exercises in which the reader is usually asked to complete certain parts of the proofs.

Overall, this is a very nice introduction to elliptic curves; although the approach is analytic, it is useful also for computer scientists interested in cryptographic applications because techniques needed for point counting (division polynomials) or for constructing curves with certain properties (complex multiplication) are discussed here.

Reviewer: Franz Lemmermeyer (Bilkent)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11G05 | Elliptic curves over global fields |

14H52 | Elliptic curves |