Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson’s biorthogonal 10W9 functions. We give an elementary construction of elliptic 6j-symbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic i...

This work was initially motivated by Miranda’s work on elliptic Weierstrass threefolds. An elliptic variety is a complex (irreducible and reduced) projective variety together with a morphism π : X → S whose fiber over a general point is a smooth elliptic curve. For example, if the elliptic fibration has a section, X can be locally expressed as a hypersurface [De]. This is called the Weierstrass...

this study concerns the existence and multiplicity of positive weak solutions for a class ofsemilinear elliptic systems with nonlinear boundary conditions. our results is depending onthe local minimization method on the nehari manifold and some variational techniques. alsoby using mountain pass lemma, we establish the existence of at least one solution withpositive energy.

We construct new elliptic solutions of the qd-algorithm. These solutions can be interpreted as elliptic solutions of the discrete-time Toda chain as well. As a by-product, we obtain new explicit orthogonal and biorthogonal polynomials in terms of the elliptic hypergeometric function 3G2(z). Their recurrence coefficients are expressed in terms of the elliptic functions. 1991 Mathematics Subject ...

Elliptic curve cryptosystems are expected to be a next standard of public-key cryptosystems. A security level of elliptic curve cryptosystems depends on a difficulty of a discrete logarithm problem on elliptic curves. The security level of a elliptic curve cryptosystem which has a public-key of 160-bit is equivalent to that of a RSA system which has a public-key of 1024-bit. We propose an ellip...

The Elliptic Curve Discrete Logarithm Problem(ECDLP) is known to be an exponential time problem except the cases of smooth curves, supersingular curves and anomalous curves. Recently, several new methods to solve ECDLP on a prime eld were proposed. All of them try to solve ECDLP on a prime eld by lifting a given elliptic curve to low rank elliptic curves de ned over the rationals. In this exten...

The security of elliptic curve cryptosystem depends on the choice of an elliptic curve on which cryptographic operations are performed. Schoof’s algorithm is used to define a secure elliptic curve, as it can compute the number of rational points on a randomly selected elliptic curve defined over a finite field. By realizing efficient combination of several improvements, such as Atkin-Elkies’s m...

To every non-singular elliptic curve (complex torus) we assign a C∗algebra Tθ = {u, v | vu = e uv} known as noncommutative torus. It is shown that morphisms of elliptic curves generate Morita equivalence of the corresponding noncommutative tori. Real number θ we call projective curvature attached to the elliptic curve. It is proved that projective curvatures of isomorphic elliptic curves are mo...