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.
This note describes the following topics:
Solving cubic equations in two variables, Group law on the
cubic curve, Theta functions, Rank two vector bundles on elliptic curves.
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.
This note explains the following topics: Elliptic Integrals, Elliptic
Functions, Periodicity of the Functions, Landen’s Transformation, Complete
Functions, Development of Elliptic Functions into Factors, Elliptic Integrals of
the Second Order, Numerical Calculations.
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.
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.
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.
Author(s): Prof.
Dipl-Ing, Dr. techn. Michael Drmota
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.
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.