We discuss abelian variety and use it to generalize the concept of elliptic curve we discussed in the class. We state some of the basic properties of abelian variety and in particular 1-dimensional abelian variety. The main result of this report is to show that elliptic curve is that the only 1-dimensional abelian variety.

Pairing-based cryptosystems have been developing very fast in the last few years. As the key primitive, pairing is also the heaviest operation in these systems. The performance of pairing affects the application of the schemes in practice. In this report, we summarise the formulas of the Tate pairing operation on elliptic curves in different coordinate systems and describe a few observations of...

A large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties. For an elliptic curve over Q, the set of rational points forms a finitely generated abelian group. The ranks of these groups, when ranging over all elliptic curves, are conjectured to be evenly distributed between rank 0 and rank 1, with higher ranks being negligible. We will de...

A new fast addition algorithm on an elliptic curve over GF(2n) using the projective coordinates with x =X/Z and y = Y/Z2 is proposed.  2000 Elsevier Science B.V. All rights reserved.

(−1)(1!2! · · · (n− 1)!) σ(nu) σ(u)n = ∣∣∣∣∣∣∣ ℘ ℘ · · · ℘ ℘ ℘ · · · ℘ .. .. . . . .. ℘ ℘ · · · ℘ ∣∣∣∣∣∣∣ (u). (0.2) Although this formula can be obtained by a limiting process from (0.1), it was found before [FS] by the paper of Kiepert [K]. If we set y(u) = 1 2℘ (u) and x(u) = ℘(u), then we have an equation y(u) = x(u)+ · · · , that is a defining equation of the elliptic curve to which the fu...

This is an exposition of some of the main features of the theory of elliptic curves and modular forms.

We derive a general expression for an interface parameter which makes possible the design of a neutral elliptic inhomogeneity when the stress field in the surrounding matrix is a polynomial function of nth order and the composite is subjected to antiplane shear deformations. © 2005 Elsevier Ltd. All rights reserved.

Denote C (d) ∆ the relative symmetric product of this family over ∆ , parameterizing effective divisors of degree d on fibres Ct for t 6= 0. We will construct a compactification H̃d of C (d) ∆ over ∆, such that H̃d has smooth total space and the fibre over t = 0 has simple normal crossing support. By studying the fibre over t = 0 of H̃d, we understand how the symmetric products of smooth curves de...

To help motivate the Weil pairing, we discuss it in the context of elliptic curves over the field of complex numbers.

This is an exposition of some of the main features of the theory of elliptic curves and modular forms.