We discuss microscopic origin of integrability in Seiberg-Witten theory, following mostly the results of [1], as well as present their certain extension and consider several explicit examples. In particular, we discuss in more detail the theory with the only switched on higher perturbation in the ultraviolet, where extra explicit formulas are obtained using bosonization and elliptic uniformizat...

We present a lower bound for the exponent of the group of rational points of an elliptic curve over a finite field. Earlier results considered finite fields Fqm where either q is fixed or m = 1 and q is prime. Here, we let both q and m vary; our estimate is explicit and does not depend on the elliptic curve.

In this paper, we examine the Lang-Trotter conjecture for elliptic curves which possess rational 3-torsion points. We prove that if one averages over all such elliptic curves then one obtains an asymptotic similar to the one predicted by Lang and Trotter.

The first part describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. The second part discusses a relationship between equivariant elliptic cohomology and representations of loop groups. The third part investigates the representation theoretic considerations which give rise to the power operations discussed in the first part.

We discuss Chapter 6 from Silverman & Tate’s Rational Points on Elliptic Curves, in which the authors outline a means of proving Kronecker’s Jugendtraum for Q and Q(i). One considers C[n], the kernel of the multiplication-by-n map on an elliptic curve C. After defining the number field Q(C[n]), we examine instances in which its Galois group over a field K is abelian. For the elliptic curve

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.