The theory of elliptic curves has been the source of new approaches to classical problems in number theory, which have also found applications in cryptography. This volume represents the proceedings of the Advanced Instructional Workshop on Algebraic Number Theory held at the HarishChandra Research Institute. The theme of the workshop was algebraic number theory with special emphasis on elliptic curves. The volume is in three parts, the first part contains articles in the field of elliptic curves, the second contains articles on modular forms. The third part presents some basics on cryptography, as well as some advanced topics. Each part contains an introduction, which, in some sense, gives the overall picture of the contents of that part. Most of the articles are presented in a selfcontained style and they give a different flavor to the subject. In some cases, the authors have chosen to include material that is already available in textbooks in order to make this volume more complete. Graduate students who want to pursue their research career in number theory will benefit from this volume. The book is suitable for graduate students and researchers in number theory and applications to cryptography. A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels. Readership Graduate students and research mathematicians interested in number theory and applications to cryptography. Table of Contents Part I. Elliptic Curves  D. S. Nagaraj  An overview
 D. S. Nagaraj and B. Sury  A quick introduction to algebraic geometry and elliptic curves
 B. Sury  Elliptic curves over finite fields
 R. Tandon  The NagellLutz theorem
 C. S. Rajan  Weak MordellWeil theorem
 D. S. Nagaraj and B. Sury  The MordellWeil theorem
 E. Ghate  Complex multiplication
 D. Prasad  The main theorem of complex multiplication
 T. N. Shorey  Approximations of algebraic numbers by rationals: A theorem of Thue
 S. D. Adhikari and D. S. Ramana  Siegel's theorem: Finiteness of integral points
 A. F. Brown  \(p\)adic theta functions and Tate curves
 D. S. Nagaraj  \(l\)adic representation attached to an elliptic curve over a number field
 C. S. Dalawat  Arithmetic on curves
Part II. Modular Forms  B. Ramakrishnan  Introduction
 P. Shastri  Elliptic functions
 M. Manickam and B. Ramakrishnan  An introduction to modular forms and Hecke operators
 C. S. Yogananda  \(I\)functions of modular forms
 T. N. Venkataramana  On the EichlerShimura congruence relation
Part III. Cryptography  A. K. Bhandari  Cryptography
 R. Thangadurai  Classical cryptosystems
 A. K. Bhandari  The public key cryptography
 A. Nongkynrih  Primality and factoring
 R. Balasubramanian  Elliptic curves and cryptography
