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 Tata Institute of Fundamental Research 2006; 400 pp; hardcover ISBN-10: 81-7319-502-1 ISBN-13: 978-81-7319-502-0 List Price: US$45 Member Price: US$36 Order Code: TIFR/9 These notes constitute a lucid introduction to "Elliptic Curves", one of the central and vigorous areas of current mathematical research. The subject has been studied from diverse viewpoints--analytic, algebraic, and arithmetical. These notes offer the reader glimpses of all three aspects and present some of the basic important theorems in all of them. The first part introduces a little of the theory of Riemann surfaces and goes on to the study of tori and their projective embeddings as cubics. This part ends with a discussion of the identification of the moduli space of complex tori with the quotient of the upper half plane by the modular groups. The second part handles the algebraic geometry of elliptic curves. It begins with a rapid introduction to some basic algebraic geometry and then focuses on elliptic curves. The Rieman-Roch theorem and the Riemann hypothesis for elliptic curves are proved, and the structure of the endomorphism ring of an elliptic curve is described. The third and last part is on the arithmetic of elliptic curves over $$Q$$. The Mordell-Weil theorem, Mazur's theorem on torsion in rational points of an elliptic curve over $$Q$$, and theorems of Thue and Siegel are among the results which are presented. There is a brief discussion of theta functions, Eisenstein series and cusp forms with an application to representation of natural numbers as sums of squares. The notes end with the formulation of the Birch and Swinnerton-Dyer conjectures. There is an additional brief chapter (Appendix C), written in July 2004 by Kirti Joshi, describing some developments since the original notes were written up in the present form in 1992. A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka. Readership Graduate students and research mathematicians interested in elliptic curves. Table of Contents Part I. Analytic theory of Elliptic Curves Doubly periodic functions Riemann surfaces Tori Isomorphism of tori and the $$j$$ invariant All smooth cubics are complex tori Moduli Part II. Geometry of Elliptic Curves Some results from commutative algebra Varieties Further properties of varieties Intersection theory for plane curves Geometry of curves Geometry of elliptic curves Structure of endomorphisms of elliptic curves Part III. Arithmetic of Elliptic Curves Rational points on curves The Mordell-Weil theorem for elliptic curves over $${\mathbf Q}$$ Computing the Mordell-Weil group Integer points, and the theorems of Thue and Siegel Representation of numbers by squares The conjecture of Birch and Swinnerton-Dyer Appendices Bibliography