Algebraic Theory of KP Equations

Elliptic Curves An Introduction

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.

Author(s): University of Limerick

99 Pages

Elliptic Curves by Zhiyuan Bai

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.

Author(s): Zhiyuan Bai

41 Pages

Elliptic Curves by Thomas Kramer

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.

Author(s): Thomas Kramer

77 Pages

Introduction to elliptic curves

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.

Author(s): Christophe Ritzenthaler

59 Pages

The elliptic modular functions associated with the elliptic norm curve E7

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.

Author(s): Roscoe Woods

106 Pages

Mathematical Foundations of Elliptic Curve Cryptography (PDF 113P)

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

113 Pages

Algebraic Theory of KP Equations

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.

Author(s): Motohico Mulase

61 Pages

Current Topics in Complex Algebraic Geometry(1995)

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.

Author(s): Herbert Clemens and Janos Kollar

NA Pages