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Elliptic curves by Miles Reid
This course note aims to give a basic overview of some of the main lines of study of elliptic curves, building on the student's knowledge of undergraduate algebra and complex analysis, and filling in background material where required (especially in number theory and geometry). Particular aims are to establish the link between doubly periodic functions, Riemann surfaces of genus 1, plane cubic curves, and associated Diophantine problems.
Author(s): Miles Reid
NA Pages 
Elliptic Curves by Zhiyuan Bai
This note covers fermats method of infinite descent, Some remarks on algebraic curves, Weierstrass equations, The group law, Isogeny, The invariant differential, Elliptic curves over finite fields, Formal groups, Elliptic curves over local fields, Elliptic curves over number fields, Kummer theory, Mordell weil theorem, Heights, Dual isogenies and weil pairing, Galois cohomology, Descent by cyclic isogeny.
Author(s): Zhiyuan Bai
41 Pages
Elliptic Curves by Pete L. Clark
An elliptic curve is an object defined over a ground field K. This PDF covers the following topics related to Elliptic Curves : What is an elliptic curve?, Mordell-Weil Groups, Background on Algebraic Varieties, The Riemann-Roch Express, Weierstrass Cubics, The l-adic Tate module, Elliptic Curves Over Finite Fields, The Mordell-Weil Theorem I: Overview, The Mordell-Weil Theorem II: Weak Mordell-Wei, The Mordell-Weil Theorem III: Height Functions, The Mordell-Weil Theorem IV: The Height Descent Theorem, The Mordell-Weil Theorem V: Finale, More On Heights, Diophantine Approximation, Siegel’s Theorems on Integral Points.
Author(s): Pete L. Clark
90 Pages
Elliptic Curves by Thomas Kramer
Elliptic curves belong to the most fundamental objects in mathematics and connect many different research areas such as number theory, algebraic geometry and complex analysis. Their definition and basic properties can be stated in an elementary way: Roughly speaking, an elliptic curve is the set of solutions to a cubic equation in two variables over a field. This PDF covers the following topics related to Elliptic Curves : Analytic theory of elliptic curves, Elliptic integrals, The topology of elliptic curves, Elliptic curves as complex tori, Complex tori as elliptic curves, Geometric form of the group law, Abel’s theorem, The j-invariant, The valence formula, Geometry of elliptic curves, Affine and projective varieties, Smoothness and tangent lines, Intersection theory for plane curves, The group law on elliptic curves, Abel’s theorem and Riemann-Roch, Weierstrass normal forms, The j-invariant, Arithmetic of elliptic curves, Rational points on elliptic curves, Reduction modulo primes and torsion points, An intermezzo on group cohomology, The weak Mordell-Weil theorem, Heights and the Mordell-Weil theorem.
Author(s): Thomas Kramer
77 Pages
Elliptic Curve Cryptography by Abhijit Das
This note explains the following topics: Arithmetic of Elliptic Curves, Classical Elliptic-Curve Cryptography, Efficient Implementation, Introduction to Pairing, Pairing-Based Cryptography, Sample Application—ECDSA Batch Verification.
Author(s): Abhijit Das
137 Pages