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.
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: 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.
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.