##
**Elliptic functions and elliptic integrals. Transl. from the orig. Russian manuscript by D. Leites.**
*(English)*
Zbl 0946.11001

Translations of Mathematical Monographs. 170. Providence, RI: American Mathematical Society (AMS). x, 185 p. (1997).

The development of the theory of elliptic integrals, elliptic functions, and elliptic curves must be seen as one of the greatest achievements of nineteenth-century mathematics. Originated by the pioneering work of Abel, Gauss, Jacobi, and Legendre in the first half of the 19th century, the subject became one of the central mathematical research topics in the sequel and, simultaneously, one of the fundaments for the developments of several modern branches of mathematics such as complex algebraic geometry, complex analysis, the theory of algebraic and Abelian function fields, diophantine geometry, and topology.

In spite of its long history marked by the famous names of Weierstrass, Poincaré, Picard, Fricke, Hurwitz, and many others, the theory of elliptic functions and elliptic curves is still a very alive and rapidly developing domain of mathematics, a seemingly inexhaustive source of conceptual ideas, general techniques, deep and significant problems, conjectures, and extensive applications as well as a fundamental toolkit in various non-classical fields of modern mathematics and mathematical physics. In particular, the developments in both mathematics (Fermat’s Last Theorem, moduli theory in algebraic and arithmetic geometry, nonlinear partial differential equations and soliton theory, etc.) and physics (quantum field theories, strings, integrable Hamiltonian systems, etc.), over the past two decades, have strikingly demonstrated the even increasing importance and ubiquity of elliptic functions and curves in the mathematical sciences as a whole.

Therefore, and despite the existence of numerous excellent textbooks on the subject, both classical and modern ones, any new, specific introduction to this fascinating topic should be highly welcome.

The book under review provides such another, indeed very special introductory text on elliptic functions and elliptic curves. Based on courses taught by the two authors between 1991 and 1993 to students at all levels in Moscow, and translated into English from the original Russian manuscripts, the text is designed for those readers who encounter this topic for the first time. As the authors promise in their preface to the book, the text does not assume from the reader any advanced knowledge beyond the limits of the basic courses in mathematics at universities, and it is really accessible to the widest range of readers, including students of mathematics and physics, school teachers, and even talented high school students.

As to the motivation, arrangement, and presentation of the material on elliptic functions and curves covered here, the authors focus on both the classical approach and the historical line of development. Stressing the elementary aspects and the vast amount of concrete classical examples from the modern point of view, with references to the most recent texts on the subject, the authors have masterly managed the rather difficult task to throw a bridge between the original work of the old masters, on the one hand, and the current developments, on the other hand.

The text consists of seven chapters, each of which is subdivided into several sections.

Chapter 1 discusses the elementary geometry of plane cubic curves, including the group law, their tangent lines and inflection points, normal forms of nonsingular cubics, and singular cubic curves.

Chapter 2 gives an introduction to elliptic functions and elliptic integrals. Starting from a brief description of the topological structure of nonsingular projective plane curves via coverings of the projective line, the authors treat the following topics: general elliptic functions, the elliptic Weierstrass function, projective embeddings of complex 1-tori as nonsingular plane cubics, elliptic integrals and their normal forms à la Legendre and Weierstrass, addition theorems for elliptic integrals, the elliptic Jacobi functions, and the Weierstrass theorem on functions admitting an algebraic addition theorem.

Chapter 3 is devoted to the study of algebraic curves in the affine plane, whose arc length can be expressed as an elliptic integral. This nice illustration of the foregoing material transpires the geometrical (and historical) origin of the subject, and culminates in the discussion of Serret’s construction of curves with elliptic arcs.

This classical spirit also characterizes the contents of Chapter 4, in which the geometry and the algebra of the lemniscate are investigated. Along the historical path marked by names of Cassini, Bernoulli, Fagnano, Euler, Abel, Gauss, Eisenstein, and others, the authors discuss the ancient problem of dividing the arc of the lemniscate into a given number of equal parts. In view of the algebraic nature of this classical problem, the construction of regular polygons and the elements of Galois theory are touched upon, and the algebraic equation for the division of the lemniscate is derived thereafter. The highlight of this chapter is Abel’s famous theorem on the divisibility of the arc of the lemniscate. The authors present two different proofs of Abel’s theorem, namely the classical one due to Eisenstein, and the elegant recent proof discovered by M. Rosen [Am. Math. Mon. 88, 387-395 (1981; Zbl 0491.14023)]. This leads to the arithmetical aspects of elliptic curves, a topic that the following Chapter 5 deals with.

More precisely, Chapter 5 is devoted to the investigation of some classical diophantine equations related to rational points on elliptic curves. At the end, this discussion culminates in presenting a proof of Mordell’s theorem on the finite generation of the abelian group of rational points on a rational elliptic curve. This chapter concludes with an outlook to the rank and the torsion group of an elliptic curve, including a few remarks on the current state of knowledge with respect to the famous conjectures of Hasse-Weil, Weil-Taniyama, Fermat, and Birch-Swinnerton-Dyer. Also, B. Mazur’s fundamental theorem on the structure of the group of torsion points of an elliptic curve defined over \(\mathbb{Q}\) is explained and illustrated by concrete examples.

The last two chapters of the book turn to the problem of solving algebraic equations of degree five.

Chapter 6 gives an introduction to the theory of algebraic equations in general. The main topics here are: symmetric polynomials, Lagrange resolvents, roots of unity, Abel’s theorem on the unsolvability in radicals of the general quintic equation, the Tschirnhaus transformation, and the Bring equation.

Chapter 7 then discusses theta functions and their application to solving quintic algebraic equations. In the course of this program, the authors describe the basic properties of the generating theta functions in one variable, their transformation behavior with respect to the lattice parameter, the absolute invariant of an elliptic curve as a modular function, and the fundamental domain for the absolute invariant.

Altogether, this introductory text on elliptic functions and elliptic integrals provides a great panoramic view for the beginner in the field. The style of writing is very cultured, with a sympathetic understanding and respect of the pioneering work of the old masters, and with a delightful sense of aesthetics and beauty. This book really breathes the charm and the fine art of the mathematics of Abel’s era, and it is mainly this particular feature that makes the book under review fairly unique and highly valuable. Experts and teachers can benefit from the reading of this text likewise.

In spite of its long history marked by the famous names of Weierstrass, Poincaré, Picard, Fricke, Hurwitz, and many others, the theory of elliptic functions and elliptic curves is still a very alive and rapidly developing domain of mathematics, a seemingly inexhaustive source of conceptual ideas, general techniques, deep and significant problems, conjectures, and extensive applications as well as a fundamental toolkit in various non-classical fields of modern mathematics and mathematical physics. In particular, the developments in both mathematics (Fermat’s Last Theorem, moduli theory in algebraic and arithmetic geometry, nonlinear partial differential equations and soliton theory, etc.) and physics (quantum field theories, strings, integrable Hamiltonian systems, etc.), over the past two decades, have strikingly demonstrated the even increasing importance and ubiquity of elliptic functions and curves in the mathematical sciences as a whole.

Therefore, and despite the existence of numerous excellent textbooks on the subject, both classical and modern ones, any new, specific introduction to this fascinating topic should be highly welcome.

The book under review provides such another, indeed very special introductory text on elliptic functions and elliptic curves. Based on courses taught by the two authors between 1991 and 1993 to students at all levels in Moscow, and translated into English from the original Russian manuscripts, the text is designed for those readers who encounter this topic for the first time. As the authors promise in their preface to the book, the text does not assume from the reader any advanced knowledge beyond the limits of the basic courses in mathematics at universities, and it is really accessible to the widest range of readers, including students of mathematics and physics, school teachers, and even talented high school students.

As to the motivation, arrangement, and presentation of the material on elliptic functions and curves covered here, the authors focus on both the classical approach and the historical line of development. Stressing the elementary aspects and the vast amount of concrete classical examples from the modern point of view, with references to the most recent texts on the subject, the authors have masterly managed the rather difficult task to throw a bridge between the original work of the old masters, on the one hand, and the current developments, on the other hand.

The text consists of seven chapters, each of which is subdivided into several sections.

Chapter 1 discusses the elementary geometry of plane cubic curves, including the group law, their tangent lines and inflection points, normal forms of nonsingular cubics, and singular cubic curves.

Chapter 2 gives an introduction to elliptic functions and elliptic integrals. Starting from a brief description of the topological structure of nonsingular projective plane curves via coverings of the projective line, the authors treat the following topics: general elliptic functions, the elliptic Weierstrass function, projective embeddings of complex 1-tori as nonsingular plane cubics, elliptic integrals and their normal forms à la Legendre and Weierstrass, addition theorems for elliptic integrals, the elliptic Jacobi functions, and the Weierstrass theorem on functions admitting an algebraic addition theorem.

Chapter 3 is devoted to the study of algebraic curves in the affine plane, whose arc length can be expressed as an elliptic integral. This nice illustration of the foregoing material transpires the geometrical (and historical) origin of the subject, and culminates in the discussion of Serret’s construction of curves with elliptic arcs.

This classical spirit also characterizes the contents of Chapter 4, in which the geometry and the algebra of the lemniscate are investigated. Along the historical path marked by names of Cassini, Bernoulli, Fagnano, Euler, Abel, Gauss, Eisenstein, and others, the authors discuss the ancient problem of dividing the arc of the lemniscate into a given number of equal parts. In view of the algebraic nature of this classical problem, the construction of regular polygons and the elements of Galois theory are touched upon, and the algebraic equation for the division of the lemniscate is derived thereafter. The highlight of this chapter is Abel’s famous theorem on the divisibility of the arc of the lemniscate. The authors present two different proofs of Abel’s theorem, namely the classical one due to Eisenstein, and the elegant recent proof discovered by M. Rosen [Am. Math. Mon. 88, 387-395 (1981; Zbl 0491.14023)]. This leads to the arithmetical aspects of elliptic curves, a topic that the following Chapter 5 deals with.

More precisely, Chapter 5 is devoted to the investigation of some classical diophantine equations related to rational points on elliptic curves. At the end, this discussion culminates in presenting a proof of Mordell’s theorem on the finite generation of the abelian group of rational points on a rational elliptic curve. This chapter concludes with an outlook to the rank and the torsion group of an elliptic curve, including a few remarks on the current state of knowledge with respect to the famous conjectures of Hasse-Weil, Weil-Taniyama, Fermat, and Birch-Swinnerton-Dyer. Also, B. Mazur’s fundamental theorem on the structure of the group of torsion points of an elliptic curve defined over \(\mathbb{Q}\) is explained and illustrated by concrete examples.

The last two chapters of the book turn to the problem of solving algebraic equations of degree five.

Chapter 6 gives an introduction to the theory of algebraic equations in general. The main topics here are: symmetric polynomials, Lagrange resolvents, roots of unity, Abel’s theorem on the unsolvability in radicals of the general quintic equation, the Tschirnhaus transformation, and the Bring equation.

Chapter 7 then discusses theta functions and their application to solving quintic algebraic equations. In the course of this program, the authors describe the basic properties of the generating theta functions in one variable, their transformation behavior with respect to the lattice parameter, the absolute invariant of an elliptic curve as a modular function, and the fundamental domain for the absolute invariant.

Altogether, this introductory text on elliptic functions and elliptic integrals provides a great panoramic view for the beginner in the field. The style of writing is very cultured, with a sympathetic understanding and respect of the pioneering work of the old masters, and with a delightful sense of aesthetics and beauty. This book really breathes the charm and the fine art of the mathematics of Abel’s era, and it is mainly this particular feature that makes the book under review fairly unique and highly valuable. Experts and teachers can benefit from the reading of this text likewise.

Reviewer: Werner Kleinert (Berlin)

### MSC:

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

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

33-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions |

11G05 | Elliptic curves over global fields |

14H52 | Elliptic curves |

33E05 | Elliptic functions and integrals |