Elliptic Curves, Modular Forms and Cryptography: Proceedings of the Advanced Instructional Workshop on Algebraic Number Theory

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Front Matter ....Pages i-viii Front Matter ....Pages 1-1 An overview (D. S. Nagaraj)....Pages 3-4 A Quick Introduction to Algebraic Geometry and Elliptic Curves (D. S. Nagaraj, B. Sury)....Pages 5-31 Elliptic Curves over Finite Fields (B. Sury)....Pages 33-47 The Nagell-Lutz Theorem (Rajat Tandon)....Pages 49-61 Weak Mordell-Weil theorem (C. S. Rajan)....Pages 63-72 The Mordell-Weil Theorem (D. S. Nagaraj, B. Sury)....Pages 73-84 Complex Multiplication (Eknath Ghate)....Pages 85-108 The Main Theorem of Complex Multiplication (Dipendra Prasad)....Pages 109-118 Approximations of Algebraic Numbers by Rationals: A Theorem of Thue (T. N. Shorey)....Pages 119-137 Siegel’s Theorem: Finiteness of Integral Points (S. D. Adhikari, D. S. Ramana)....Pages 139-149 p-adic Theta Functions and Tate Curves (Alexander F. Brown)....Pages 151-165 ℓ-adic Representation Attached to an Elliptic Curve over a Number Field (D. S. Nagaraj)....Pages 167-191 Arithmetic on Curves (Chandan Singh Dalawat)....Pages 193-200 Front Matter ....Pages 201-201 Introduction (B. Ramakrishnan)....Pages 203-204 Elliptic Functions (Parvati Shastri)....Pages 205-221 An Introduction to Modular Forms and Hecke Operators (M. Manickam, B. Ramakrishnan)....Pages 223-245 L-Functions of modular forms (C. S. Yogananda)....Pages 247-254 On the Eichler-Shimura Congruence Relation (T. N. Venkataramana)....Pages 255-265 Front Matter ....Pages 267-267 Cryptography (Ashwani K. Bhandari)....Pages 269-274 Classical Cryptosystems (R. Thangadurai)....Pages 275-286 The Public Key Cryptography (Ashwani K. Bhandari)....Pages 287-301 Primality and Factoring (Amora Nongkynrih)....Pages 303-323 Elliptic Curves and Cryptography (R. Balasubramanian)....Pages 325-345