

This section contains free ebooks 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 And Modular FormsRobert C. RhoadesPDF  142 Pages  EnglishThis note is an introduction to
elliptic curves and modular forms. These play a central
role in modern arithmetical geometry and even in applications to cryptography.
On the elliptic curve side, we shall cover elliptic curves over finite fields,
over the complex numbers, and over the rationals. Topics covered includes:
Congruent Numbers, Hasse Principal, Elliptic Curves over C, Riemann Surfaces,
Addition Law, Explicit Group Law, Modular Forms and Modular Curves, Elliptic
Curves over Finite Fields, Dual Isogeny, Dichotomy Between SuperSingular and
Ordinary Case, Mordell’s Theorem.
 Elliptic Curves by J.S. MilneJ.S. MilnePDF  163 Pages  EnglishThis
note explains the following topics: Plane Curves, Rational Points on
Plane Curves, The Group Law on a Cubic Curve, Functions on Algebraic Curves and
the RiemannRoch Theorem, Reduction of an Elliptic Curve Modulo p, Elliptic
Curves over Qp, Torsion Points, Neron Models, Elliptic Curves over the Complex
Numbers, The MordellWeil Theorem: Statement and Strategy, The TateShafarevich
Group; Failure Of The Hasse Principle, Elliptic Curves Over Finite Fields, The
Conjecture of Birch and SwinnertonDyer, Elliptic Curves and Sphere Packings,
The Conjecture of Birch and SwinnertonDyer, Algorithms for Elliptic Curves.
 Elliptic Curve HandbookIan ConnellPDF  327 Pages  EnglishThis
book covers the following topics: Projective coordinates,
Cubic to Weierstrass, Formal Groups, The MordellWeil theorem, Twists, Minimal
Weierstrass Equations, Isomorphisms of elliptic curves , Automorphisms and
fields of definition, Kraus’s theorem.
 Introduction to elliptic curvesChristophe RitzenthalerPDF  59 Pages  EnglishThis 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.
  Elliptic Curves by David LoefflerDavid LoefflerPDF  74 Pages  EnglishThis note provides the
explanation about the following topics: Definitions and Weierstrass equations,
The Group Law on an Elliptic Curve, Heights and the MordellWeil Theorem, The
curve, Completion of the proof of MordellWeil, Examples of rank calculations,
Introduction to the Padic numbers, Motivation, Formal groups, Points of finite
order, Minimal Weierstrass Equations, Reduction mod pII and torsion points over
algebraic extensions, Isogenies, Hasse’s Theorem and Galois cohomology.
 Elliptic curves, L functions, and CM pointsShouWu ZhangPDF  64 Pages  EnglishThe 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, Lfunctions and modular forms, Complex
multiplications, Shimura curves, CMpoints and Heegner points, Lfunctions with
characters and CMpoints with characters.
 An Introduction to the Theory of Elliptic Curves (PDF 104P)Joseph
H. SilvermanPDF  104 Pages  EnglishCovered 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, LSeries,
BirchSwinnertonDyer.
 Mathematical Foundations of Elliptic Curve Cryptography (PDF 113P)Prof.
DiplIng, Dr. techn. Michael DrmotaPDF  113 Pages  EnglishThis 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 topologyMatthew
AndoPDF  23 Pages  EnglishThis note covers the following topics: Geometric reformulation, The AdamsNovikov spectral sequence,
Elliptic cohomology, What is TMF, Geometric and Physical Aspect.
 Algebraic Theory of KP EquationsMotohico
MulasePDF  61 Pages  EnglishThis note covers the following topics: The KP equation and elliptic
functions, The spectral curve of a differential operator, Grassmannians and the
geometric inverse scattering, Isospectral deformations and the KP system,
Jacobian varieties as moduli of isospectral deformations, Morphisms of curves,
Prym varieties and commuting partial differential operators.
 Current Topics in Complex Algebraic Geometry(1995)
Herbert Clemens and Janos KollarOnline  NA Pages  EnglishThis 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 GroupsJ. Lubin, J.P. Serre and J. TateOnline  NA Pages  EnglishThis note explains
many topics related to Elliptic Curves and Formal Groups.
 Elliptic curves by Miles Reid
Miles ReidOnline  NA Pages  EnglishThis 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 








