Elliptic Curves

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; It offers the reader glimpses of all three aspects and presents 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 Riemann-Roch theorem and the Riemann hypothesis for elliptic curves are proved and the structure of endomorphism ring of an elliptic curve are 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 $Q$, 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.