Park City lectures on elliptic curves over function fields

Lectures on Moduli Spaces of Elliptic Curves

The goal of these notes is to introduce orbifolds and stacks through the example of elliptic curves. This note contains below topics, Elliptic Curves and the Moduli Problem, Families of Elliptic Curves and the Universal Curve, The Orbifold and Modular Forms.

Author(s): Richard Hain

80 Pages

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, Abels theorem, The jinvariant, 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, Abels theorem and RiemannRoch, Weierstrass normal forms, The jinvariant, Arithmetic of elliptic curves, Rational points on elliptic curves, Reduction modulo primes and torsion points, An intermezzo on group cohomology, The weak MordellWeil theorem, Heights and the MordellWeil theorem.

Author(s): Thomas Kramer

77 Pages

Elliptic Functions An Elementary Text Book for Students of Mathematics

This note explains the following topics, Elliptic Integrals, Elliptic Functions, Periodicity of the Functions, Landens Transformation, Complete Functions, Development of Elliptic Functions into Factors, Elliptic Integrals of the Second Order, Numerical Calculations.

Author(s): Arthur L. Bake

147 Pages

Elliptic Curve Cryptography by Christian Wuthrich

Ellipticcurve cryptography is an approach to publickey cryptography based on the algebraic structure of elliptic curves over finite fields.

Author(s): Christian Wuthrich

98 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

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