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This section contains free e-books and guides on Elliptic Curves, some of the resources in this section can be viewed online and some of them can be downloaded.

Elliptic Curves Books
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Elliptic Curves Lecture Notes

Online | NA Pages | English
This graduate-level course note is a computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics covered includes: The Group Law, Weierstrass and Edwards Equations, Finite Fields and Integer Arithmetic, Finite Field Arithmetic, Isogenies, Endomorphism Rings, Hasse's Theorem, Point Counting, Schoof's Algorithm, Generic Algorithms for Discrete Logarithms, Elliptic Curve Primality Proving.

 
Elliptic Curve Handbook

PDF | 327 Pages | English
This book covers the following topics: Projective coordinates, Cubic to Weierstrass, Formal Groups, The Mordell-Weil theorem, Twists, Minimal Weierstrass Equations, Isomorphisms of elliptic curves , Automorphisms and fields of definition, Krausís theorem.

 
Introduction to elliptic curves

PDF | 59 Pages | English
This book covers the following topics: The group law, Elliptic curves over finite fields, Pairings, Travaux Diriges, Elliptic curves over finite fields, Number of points on elliptic curves over finite fields: theory and practice.

 
The elliptic modular functions associated with the elliptic norm curve E7

Online | 106 Pages | English
This note covers the following topics: The Groups Connected With E7, The Quadrigs On E7, The Interpretation Of The Forms F and F2, The Loci In S3.

 
Elliptic Curves by David Loeffler

PDF | 74 Pages | English
This note provides the explanation about the following topics: Definitions and Weierstrass equations, The Group Law on an Elliptic Curve, Heights and the Mordell-Weil Theorem, The curve, Completion of the proof of Mordell-Weil, Examples of rank calculations, Introduction to the P-adic numbers, Motivation, Formal groups, Points of finite order, Minimal Weierstrass Equations, Reduction mod pII and torsion points over algebraic extensions, Isogenies, Hasseís Theorem and Galois cohomology.

 
Elliptic curves, L functions, and CM points

PDF | 64 Pages | English
The aim of this note is to give a survey on recent development of the Gross Zagier formulas and their applications. Covered topics are Elliptic curves: geometry and arithmetic, L-functions and modular forms, Complex multiplications, Shimura curves, CM-points and Heegner points, L-functions with characters and CM-points with characters.

 
An Introduction to the Theory of Elliptic Curves (PDF 104P)

PDF | 104 Pages | English
Covered topics are: Elliptic Curves, The Geometry of Elliptic Curves, The Algebra of Elliptic Curves, Elliptic Curves Over Finite Fields, The Elliptic Curve Discrete Logarithm Problem, Height Functions, Canonical Heights on Elliptic Curves, Factorization Using Elliptic Curves, L-Series, Birch-Swinnerton-Dyer.

 
Mathematical Foundations of Elliptic Curve Cryptography (PDF 113P)

PDF | 113 Pages | English
This note covers the following topics: algebraic curves, elliptic curves, elliptic curves over special fields , more on elliptic divisibility sequences and elliptic nets , elliptic curve cryptography , computational aspects , elliptic curve discrete logarithm.

 
Modular curves
Elliptic curves and algebraic topology

PDF | 23 Pages | English
This note covers the following topics: Geometric reformulation, The Adams-Novikov spectral sequence, Elliptic cohomology, What is TMF, Geometric and Physical Aspect.

 
Algebraic Theory of KP Equations

PDF | 61 Pages | English
This note covers the following topics: The KP equation and elliptic functions, The spectral curve of a differential operator, Grassmannians and the geometric inverse scattering, Iso-spectral deformations and the KP system, Jacobian varieties as moduli of iso-spectral deformations, Morphisms of curves, Prym varieties and commuting partial differential operators.

 
INTRODUCTION TO ALGEBRAIC TOPOLOGY AND ALGEBRAIC GEOMETRY

PDF | 138 Pages | English
This note provides an introduction to algebraic geometry for students with an education in theoretical physics, to help them to master the basic algebraic geometric tools necessary for doing research in algebraically integrable systems and in the geometry of quantum eld theory and string theory. Covered topics are: Algebraic Topology, Singular homology theory, Introduction to sheaves and their cohomology, Introduction to algebraic geometry, Complex manifolds and vector bundles, Algebraic curves.

 
Current Topics in Complex Algebraic Geometry(1995)

Online | NA Pages | English
This note covers the following topics: Fundamental Groups of Smooth Projective Varieties, Vector Bundles on Curves and Generalized Theta Functions: Recent Results and Open Problems, The Schottky Problem, Spectral Covers, Torelli Groups and Geometry of Moduli Spaces of Curves.

 
Elliptic Curves and Formal Groups

Online | NA Pages | English
This note explains many topics related to Elliptic Curves and Formal Groups.

 
Elliptic curves by Miles Reid

Online | NA Pages | English
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.

 
Elliptic Curves by Jim Milne
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