This note explains the following
topics: Plane curves, Projective space and homogenisation, Rational points on
curves, Bachet-Mordell equation, Congruent number curves, Elliptic curves and
group law, Integer Factorization Using Elliptic Curves, Isomorphisms and
j-invariant, Elliptic curves over C, Endomorphisms of elliptic curves, Elliptic
Curves over finite fields, The Mordell–Weil Theorem, Elliptic Curve
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 Riemann-Roch Theorem, Reduction of an Elliptic Curve Modulo p, Elliptic
Curves over Qp, Torsion Points, Neron Models, Elliptic Curves over the Complex
Numbers, The Mordell-Weil Theorem: Statement and Strategy, The Tate-Shafarevich
Group; Failure Of The Hasse Principle, Elliptic Curves Over Finite Fields, The
Conjecture of Birch and Swinnerton-Dyer, Elliptic Curves and Sphere Packings,
The Conjecture of Birch and Swinnerton-Dyer, Algorithms for Elliptic Curves.
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, Kraus’s theorem.
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,
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
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.